Yo Mike,

My first harmony / music theory teacher told us
to build chords very much as you describe, with
a "manual" twist, like so:

"Pick a note you want to build a chord on".

Start counting notes on your thumb, one note for
each digit.

For example "A" and the thumb, then "B", "C",
and so on.

Now, keep the first note, but leave out every
other digit.

That leaves "A", "C", and "E".

That gives you notes that are all a third apart
from each other, as in "A" = 1, "B" = 2, then "C"
is the third note from where you started ("A")."
Same if you start on "C".

It also happens to be the definition of an "A" chord,

A,C,E = "A minor"

All of us high school frosh in the "Boys Chorus"
would be furiously counting on our fingers during
the first few tests. But soon, we found ourselves
just instantly saying chords, like "ACE", "CEG",
etc., without even having to think about it.

That is a handy skill.

So, there are a few other little "gotchas", such
as the difference between a minor / major third,
but the 1,3,5 rule is always still true.

Now we are finally caught up to where you left off.

The fun begins as you keep adding thirds, using the
same basic principals, as if we had more digits on
our extremities.

So, "jazz" chords get 1,3,5,7,9 and so on, which is
no different than just stacking those memorized
triads* one after the other ("A,C,E,G,B").

* triad is another name for a basic three note chord.

You can just keep going, but us guitarists have to
economize, on account of having not many more
strings than digits. Any very "sophisticated" jazz
chord, as played on a guitar, is bound to have some
"missing" notes. But that's part of the fun, figuring
out which ones to keep, and which ones to skip, and
learning how that effects the sound as a whole.

That's what one of my later music theory teachers
called "hearing chords", rather than individual notes.

I think guitarists, pianists, and harpists have a great
advantage in that respect. Playing chords is just part
of the nature of those instruments.

Thank you Tom Preble, hope you're doing well!

Fun stuff!
Emmit Sycamore