Chords in western music are said to be built on intervals of 3rds....
Well, this ignores developments in harmony dating back at least 100 years and with roots all the way back in the middle ages when perfect fourths were considered consonances. Quartal and quintal harmony are quite common today. You can think of triads as being built in thirds, and you can think of seventh chords and extended seventh chords as being build in thirds too. But this approach can be misleading since in practice chord voicings are what matters. In practice, in some circumstances, an extended seventh chord may be voiced without one or several of the notes derived from the stack-of-thirds construction, or the chord may be voiced in one of many inversions, carrying the interval relationships far from the original stack-of-thirds.
This chord can be said to be consonant from a basis of being harmonically pleasant.
That sounds fine; it is one way to define consonance. Others try to define consonance in terms of harmonic ratios. The first approach ("harmonically pleasant") would seem to have the advantage of being ambiguous; intervals that were once considered dissonant have come to be considered consonant, and the perfect fourth was considered consonant, then dissonant, then consonant again!
These two ways of defining dissonance seem to be talking about different things. Maybe we need better terminology here. When I was young I liked sweet foods and didn't like bitter foods; I thought that only sweet foods were delicious. When I was young I liked consonant harmonies and didn't like dissonant harmonies; I thought that only consonant harmonies were pleasing. Now that I am older, I think that bitter foods can also be delicious, and dissonant harmonies can also be pleasing. Maybe the harmonic ratio people have a point, if we could only stop using consonant to mean "pleasing". I guess I'll have to remain uncommitted on this issue.
What is the scientific process behind the selection of intervals in creating a chord?
There isn't one. You can discover one, if you like. Any such process will be much more complicated than coming to grips with thirds, though. Actual chord voicings are complex, may omit notes, may omit the root note, may emphasize dissonant intervals that can be had through inversion or added tones.
As someone who has played and continues to play a lot of chords, I can tell you that I never think anything like "this chord needs to be more dissonant, I'll change it like so to effect that." Different intervals have distinct sounds; you get to know those sounds and combine them as sounds, not as principles. Sometimes the intervals arise as if by magic, because there is a melody line or other linear structure that suggests them. You get to know the sounds of larger chord structures, i.e., specific chord voicings, and alter them to suit an occasion. You learn what things sound like, and you learn to hear better. This is what I meant in an earlier comment when I said "it usually seems better to approach it from a more phenomenological perspective." If you approach music as an exercise in scientific principles (which may or may not exist), you will miss nearly everything important about music. You are creating sounds to be experienced; you had better experience them before anyone else.
Ultimately, how are chords built through the use of intervals, what are the mechanics?
I'm not quite sure what the question is here. There is no mechanics of chord construction other than that you combine some notes. You can build a chord any way you like; you can combine any notes you like. The question is, does a combination of notes sound "good?" That is a question only you can answer by developing your own harmonic conception. Such things don't lend themselves well to words; you have to get up to your elbows in the music.
Of course, there are systematic ways to construct chords (although I'd rather think about developing harmonic knowledge; the term chord tends to imply mashing a bunch of notes together to a lot of people, and harmony is more subtle than this).
You can construct close-voiced chords in thirds, fourths, or fifths, and find all of the inversions. There are systematic ways to approach inversions too: start with drop-2 and drop-3 voicings, then learn about other drop voicing patterns. This will give you a large vocabulary of chord sounds, and you will discover that different voicings of, say a Maj7 chord, have distinctly different qualities.
You can construct chords by taking a voicing that you already know, and adding or altering one or more notes. This is very convenient in practice, and can dramatically expand your harmonic vocabulary.
You can try to enumerate every possiblity on a particular instrument and analyze those chords. Obviously, this won't be too useful if you are composing in a DAW. George van Eps did something like this with his famous (and enormous) three-volume series Harmonic Mechanisms for Guitar. The point of that series is to inquire into the harmonic devices can be played on a guitar, the sounds of those devices, and methods for getting those devices under a player's hands. Ted Greene also did something like this with Chord Chemistry, which had the goal of enumerating nearly every chord form that is playable on the guitar.
The closest I ever come to "using intervals to build chords" is when I play, say a Maj7 chord and think to myself, "there is a minor 2nd available here, and I'd like the sound of that right now." That kind of thinking might happen more or less explicitly for me when I am arranging something, probably more subconciously when I am performing something. When it does happen, I'll add a minor 2nd by adding to or altering a voicing I already know if I don't already have a pat voicing at hand. I can do this because I know lots of voicings, and I know lots of voicings because I started with a few voicings and built up a large vocabulary by doing exactly what I have been describing (starting with a vocabulary of simple voicings, altering them as needed, and adding any interesting discoveries found along the way to your vocabulary). This is pretty much how it seems to work for everyone.
I just glanced at my copy of Chord Chemistry and saw this on the very first page:
...although learning many nice inversions or different ways to play
the same chord is essential to a good guitarist, by itself it is
nothing. It is far better to know only a few nice chords and know how
to use them than to know thousands of chords without knowing where to
put them or how they relate to each other.... Some chords that are not too pleasing by themselves are excellent when used with the right combination of other chords. [Emphasis in the original]