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Why is it important to stay within the key, i.e., to use diatonic chords (minus modulation and key changes)?
My intuition is that if the song is supposed to be happy, ergo if it's to use major chords, then all chords would end up being major.
What's the deeper reality that has made diatonic chords such a staple in music theory, even if some of those chords are diminished or minor?
(I've edited out "the thing to look for" in favor of "such a staple in music theory".)
Why is it important to stay within the key
It isn't, necessarily!
However, doing so is likely to help your music sound more stable. The fundamental frequencies of important notes within a key have a strong harmonic relationship with the tonic note of the key - for example, in a Major key, some of these ratios are 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third); In a minor key, we have the minor third (6:5), which is also a simple relationship.
In nature/physics, a single resonating body often tends to have overtones that are integer multiples of a fundamental frequency, and therefore tend to have simple frequency relationships with each other. In music, using notes that have simple frequency relationships with the root makes the overall mixture of harmonics in the piece seem more like a single resonating body from a mathematical point of view; subjectively, it keeps the music sounding stable, and gives it an overall sense of being based around, or "wanting to return to", the root.
But is that a good thing? That sense of stability might not be what you want - and in any case, using some notes that clash with the root doesn't necessarily disrupt the sense of stability in an unpleasant way. Even in the major key, for example, the seventh clashes quite strongly with the root (although it is the major third of the fifth of the root - so it has a strong relationship 'by association').
My intuition is that if the song is supposed to be happy, ergo if it's to use major chords, then all chords would end up being major.
Some pieces, or at least parts of them, do exactly that - listen to the descending suspended and major chords in the main riff from Pinball Wizard from Tommy by The Who, or the chorus from the Timewarp. For a song that only uses minor chords, try Inner City - Good Life.
It can sound great. However, because it can mean using more notes that have a less strong relationship with the root than they have with other notes, it can cause the music to sound like it's 'jumping around' - which might be a bit too dramatic to do throughout a piece.
What's the deeper reality that has made diatonic chords be the thing to look for
If there is a deep reality, it's about maintaining the sense of stability that comes from using notes with simple frequency relationships with the root. Historically this was more important, as the 12-tone equal temperament system that we have today (which softens the clashes that we get from moving to different keys) wasn't in widespread use.
However, please don't be left with the impression that diatonic chords are "the thing to look for". They are probably the thing to look for if you want a sense of stability in your music, but if you want to add a bit of excitement, you may want to move outside of the notes from the key - whether that's "borrowing" a chord, using secondary dominants, working in one or more real modulations into your piece, or just using a non-diatonic passing note.
So for music where you have a clear major tonality, using the major versions of the ii, iii and viio chords would introduce unstable notes that are a semi-tone away from the first fourth and fifth scale degrees. These scale degrees, and their chords, are really the key to establishing tonality.
Walking through it...
I, IV, V -- all good there, and they are the most important chords in establishing tonality.
The next most commonly used chords are ii and vi. If we tried II, in C we'd have D F# A. Note how the F# is a semitone away from F (and G), this is very unstable with respect to a C tonality (but is a leading tone for G, hence secondary dominants...). Similarly VI, A C# E in C, has a C# in it, again, a tone that is very foreign to a C tonality.
Moving onto iii, in vii diminished, making them major would again result in odd sounding notes (w.r.t. a C major tonality), G# and D# and F# respectively.
Why is it important to stay within the key, i.e., to use diatonic chords (minus modulation and key changes)?
Because of the thing that you don't see, because it's not marked on the keys of the piano, or in the chord's name: the tonal center and other aspects of the harmonic context. The context is formed in your mind like lines drawn on water, when you listen to music. It stays for awhile, but it needs to be maintained. If the music stops, the sense of harmonic context gradually fades away.
You see, chords do not float in space by themselves like stand-alone universes. And the notes of a chord are not interpreted solely in relation to the chord's root note. All of them are interpreted in relation to a tonal center, when heard in an actual context of a song.
If you play a major chord by itself, you might be fooled to think that the chord is inherently "happy" in any and all possible situations where it can occur. Wrong. What you actually did by playing the chord alone was, you established a tonal center to the root note of the chord. However, if you play a chord in an actual context, where a tonal center is established somewhere else than the chord's root, then the feeling is different. More knowledgeable people can probably list other factors besides the tonal center, but the point is, chords occur in different contexts, not by themselves.
Let's take an example chord progression: Am - Dm - E - Am, the tonal center is A. If you play the chords at normal speed like in a song, the E certainly does not sound happy. It's more like the saddest part of the whole thing. But if you cut out only the part of the audio that has the E major, and keep repeating only that part for a long time, then you get a tonal center at E, and it starts to sound happy.
The word "diatonic" in the question is somewhat irrelevant.
If you take a single KEY, e.g. Cmaj (CDEFGAB - only THESE seven notes!) you get 7 diatonic chords C, Dm, Em, F, G, Am, Bdim - these 7 chords contain only these 7 notes CDEFGAB.
Hence, this so called "stability". If you use only these chords you are technically using only C key notes. But some of these chords are MINOR, some major, one is dim.
Strictly speaking, if you use A chord in the key of C you are temporarily out of those 7 notes. You see A contains C#, so if you use A you get C# note which is (formally) not in the key of C. So very formally speaking Am is in the key of C, while A is not.
Thus, it's not always OK to use only major or only minor chords. This kind of thinking could be possible with simple blues (major blues with only major chords, minor with minor, but it's just for simple cases).
My intuition is that if the song is supposed to be happy,
this impression is subjective (when ever you'll statistical evidence that minor sounds sad. do you find, that the famous "a la turca" by Mozart doesn't sound happy?
why not use only minor chords?
in a minor scale the chords of III and the VI are major chords. you would be very limited just using only minor chords or only major chords, as already mentioned the dominant.
another example is the secondary dominant to the IV
If a tune changes in the chorus to the predominant IV it is often introduced by a major7 based on the final root chord transformed to a V7
e.g. am -> A7 = (V7) -> dm
A few answers get into repeating a description of common practice harmony.
Another brings in some acoustics.
But, in your question you use the word 'intuition'...
My intuition is that if the song is supposed to be happy, ergo if it's to use major chords, then all chords would end up being major.
...let's try a non-technical, emotional perspective to look for an answer.
If using only major chords was musically using only the happy chords, then on an emotional level what would this be expressing?
Has any human being even, in all of human history, even been only happy?
If you could hypothetically remove all unhappiness, would a person know what happiness is? How could someone describe happiness without experiencing unhappiness?
What's the deeper reality...?
Only happy things is not realistic. It's delusional.
Musically you could try to represent a kind of delusional, happy-land, and it would make sense for it to sound not normal. Using only major chords could be a approach to take. The result will surely sound unusual. Look up chromatic mediants it's a special kind of relationship between two major chords, but it's effect is typically described as some kind of emotional disturbance not happy!
Of course, in normal human experience, happiness is just part of the whole emotional spectrum. Tonal music flows between various chord qualities just like a real person experiences a whole range of emotions. Some tonal music shifts the emphasis between major and minor/diminished sounds to express happy and unhappy, but you should keep in mind that a lot of that is cultural. That emotional dichotomy isn't found in all musical traditions.
Finally, I would focus only on harmony as a means to express happiness. Tempo, rhythm, and dynamics are important too. Personally, I don't think those aspects are mere expressive shadings. If you took some music which harmonically was major key, dynamically moderate, with simple metrical rhythms, and then transformed the non-harmonic aspects - doubled the tempo, add crescendo/sforzando, use syncopation perhaps polyrhythms, then the music could become 'giddy' or 'frenetic.' Maybe is not necessarily un-happy, but that will bring us back around to ask: what is happiness?
Using all major chords in a passage with a melody that moves in stepwise motion often results in your piece barely--or perhaps even not--adhering to any key. Considerable stretches of both Gustav Holst's "Mars" from The Planets and Alberto Ginastera's "Danza del gaucho matrero, Op. 2, No. 3" from his Danzas Argentinas use exclusively major chords, often in stepwise motion. While "Danza del gaucho matrero" still manages to sound jovial (if strange) in those sections, "Mars" sounds tense and alien in its major-chords-only sections.
And those major-chords-only sections sound tense and strange because they have very loose senses of key. Take the D♭-C-B-C-D♭-E♭-F-G-A♭-G chord progression in "Mars", for example. In the space of two measures, these chords use every note in the chromatic scale. That's definitely not a recipe for having this music sound like it's in a diatonic key.
While you can get away with using only major chords in a happy, major-key work as long as you restrict yourself (I'd say I, IV, V, and V/V are safe), unrestricted use of only major chords has a tendency to make music sound strange instead of happy because of its atonal tendencies.
Simple theory: for a major scale (only) the I is major, the ii and iii are minor, the IV and V are major, the vi is minor and 7th is diminished or half diminished.
This is fixed music theory because chords are derived from scales by stacking thirds.
A composer can do whatever they want to write a song. It is a creative process. BUT there are a lot of “rules” people use so that the song fits a genre. Diatonic music uses diatonic scales and chords. There is non-diatonic music but it is very different from what most people consider music.
