Why are dyads not the basis or even considered a chord by a lot of music theorists?
They certainly used to be. The modern theory of chords evolved gradually over several hundred years during which the primary theoretical framework was dyadic. Music in three or more parts that modern ears would recognize as chords had already existed for a few centuries before theorists started talking about chords as such.
That some theorists distinguish dyads from chords is a bit arbitrary, but it does reflect the harmonic ambiguity of dyads. If the interval is perfect, the harmony could be major or minor (for example, to C and G you could add either E or E♭), and if the interval is major or minor, you can add a perfect interval related to either element (for example, to C and E you can add G or A; to E and G you can add B or C). The dyad is therefore somewhat problematic in harmonic analysis, and the theorists who exclude it from the definition of "chord" are mostly theorizing about harmonic analysis.
Why are tetrads (pentads etc), only considered extensions of the triad?
The historical perspective is also helpful to consider here. Until maybe 100 years ago, it was highly irregular to end with anything more than three distinct pitch classes. Such chords were considered dissonant because they contained at least one second (or seventh). The dissonance was not only an acoustic property but a contrapuntal one: it had to be resolved, so it couldn't be present in the final chord. When Rameau developed his theory that equates the chord EGC with CEG, this led to a chordal theory based on the two possible triads that contain a perfect fifth, the major and minor triads.
Much later, people started regarding "extra" tones in chords as color. If you add a sixth or seventh to a major triad, you no longer need to hear it as tension requiring resolution; it could just be a different sonority that is still restful. In this context, however, the underlying harmonic relationships can still be described in terms of triads, which leads us to refer to the "extra" tones as "color."
Surely someone will think of an exception to the preceding paragraph, but to illustrate what I mean, consider the chord progression Em7-Am7-Dm7-G7-C7. Any song using this very tonal circle-of-fifths progression can also be played without any of the sevenths: Em-Am-Dm-G-C. The harmonic functions of the chords do not change, but the color is very different.
People say extension’s and add chords only provide more “color” or “dissonance”. By color I don’t fully understand what they mean, I do understand what added dissonance would mean though, based on the harmonic series. With 3 notes you can have all consonant pitches, (Root, P5, M3/M6) or even (Root, P5, P4 /I like sus chords/). But once you get to 4 notes you have to add the 2nd or 7th which are more dissonant. Is this main reason Triads are as far as you can go before beginning to add dissonance to a consonant combination of pitches?
This is more or less it. The thing about "color" is, as I've said above, a change in the role of the acoustic dissonance: where before it provided from contrapuntal tension (also called "dissonance," which is sometimes confusing), it now serves to give the chord a different sonority.
Basically, are Tetrads the beginning of where we start to add “dissonant” pitches?
Remember that the perfect fourth is frequently said to be consonant or dissonant depending on context. Also consider the diminished fifth/augmented fourth, which is always dissonant whether there is a third tone (or more) added to it.
Lastly, usually triads are played and accompanied by vocals (a 4th pitch) which is basically extended and added harmony. Meaning, the triads are definitely the harmonic foundation, leaving the voice to play around with consonance and dissonance on top of the triads. Do I have this correct also?
In some styles, yes. But more generally, the solo melody (whether a voice or an instrument) will basically have one of the chord tones, too. In fact, you will often see songs where a chord symbol seems to have been chosen specifically to reflect the fact that the melody has a certain pitch. It's fairly rare that the melody's main pitch at any point is not reflected in the chords. For example, you're not likely to see a G7 with the melody on A; instead, the chord symbol would be some variety of G9.
The melody might have some figures that use non-chord tones ("play around with consonance and dissonance") but in an ensemble context any part can do that. For example, if the chord is C then the melody can have C-D-E without the chord having to change. But an inner part could also have E-F-G at the same time. If the passing tones (D and F) happen quickly enough then we would not analyze that as a change of chord.
Even with a lead sheet, depending on the degree of precision indicated by the given chords, a pianist or guitarist can play around with consonance and dissonance, for example seeing C and playing C-Csus4-C or adding a little scale or some other lick, such as the passing tones mentioned in the previous paragraph. Similarly, a good bassist can embellish the indicated bass notes with scales and neighbor tones. The chords just specify the harmonic framework. They aren't a precise indication of everything anyone should be playing. A composer or arranger who wants that degree of precision writes a score in staff notation, not a lead sheet.