What you are missing I think is the main reference; to the root.
Also, I think triads make sense to you, so you might be trying to make too much out of that connection- yes - they are everywhere. Leveraging those relationships is important, and powerful but itās a side effect of how chords happen via stacked 3rds. Whatās a 5th? Why itās 2 3rdsā¦ whatās a 7th? 3 3rds. Thatās a mouthful to say though isnāt it? How does it sound? Happy, sad orā¦ different. Happy = Major, minor = sad, and different = dominant. From there, the flavours go on to varying degrees of sadness etc how can we communicate this? We need a reference point. A root.
The root gives R 3 5 7, which are all tertially stacked 3rds (major and minor). Thatās 4 notes. A scale has 7, so thereās only 3 left. Available tensions for each chord? 9,11,13 which is basically a 2,4,6 just in the next octave because if itās in the same octaveā¦ you have a scale.
So if you keep stacking 3rds diatonically you will eventually have aā¦ scale. Actually in one chord you could have every chord diatonic to that original chord eventually stacked over top of that one.
Soā¦ now you know that one can find not just 2 triads in a chord, but 7+ if you keep going.
Thatās inefficient, and cumbersome. Simplify, by giving it a name. The key intervals in a chord in relation to the root - is the 3rd and the 7th. No matter what else is going on, THOSE 2 are the ones that matter. 3rd tells you what it is, and the 7th tells you where itās going; you have a choice of 2 for both, and the combinations yield the following fundamental chords. Diatonic to major scale harmony.
Maj7
Min7
Dom7
This ties in with knowing your intervals, key signatures, chord construction, modality and harmonization
Some other 7th chords that are just as important but applicable to melodic minor or harmonic minor harmony or have other altered intervals besides 3rds and 7ths or perhaps have a 3rd/7th that has been substituted with a different interval.
Min7b5
Min/maj7
Dom7sus4
And from there,
Maj6, Min6 diminished 7th etc etc
Even with triads thereās Aug, dim, sus4, sus2 etc.
Hopefully that helps a bit - itās hard to explain one thing without having to explain a lot of things, and an Internet forum isnāt the best medium to explain this lol
Apologies as I am certain I left some things out, and probably just confused you. I think you are missing/misunderstanding some of the basics and some of those gaps leave you asking questionsā¦
My suggestion is to take a long hard look at the fundamentals; theyāve been around for so very long and the basics are kind of the constants even if the school of thought is different.
Notation, and written music fits in here as well.
Good luck!