Good evening, @cyman1964ukg, and ...
... Plenty to read and amuse you here, and lots to learn and share.
Simon...
It's actually a lot more simple than you're making it, and you're very nearly 'there' with your current reflections in chord construction. The only issue is that you're looking for a 'mathematical' 'logic' that just is not there (and doesn't need to be...). Your starting point of creating with 'triads' (alternate notes of the scale...) works well for Major and Minor chords, and only needs extending a touch by adding the next 'alternate' note to the triad to get the Major 7th and Minor 7th chords. It's easy to visualise on a piano keyboard, adding the 'B' to the 'C' chord (the one below the octave 'C'...) creates the CMaj7 chord. This is consistant when doing the same to the 'DMin' chord (add the next alternate note, ie: the octave 'C'...) and you're crated the DMin7 chord. Continue for each diatonic note of the 'C Major' scale, using alternate notes from the scale, and you've got most of the chords needed for many, many songs. If we 'break' this chord system, and use a flattened '7th' (ie: 'Bb' instead of 'B'...) we get a chord which, when played, has a sonic tendency to want to 'resolve' to the CMajor, which is why it takes the name 'Dominant'. That Bb is not in the C major scale, but as it's a very powerful sounding chord, it has it's place in so many songs, from cowboy campfire songs to much early Bob Dylan stuff. It's a staple in the chord Pantheon. However, the chord naming of 'C7' tricks us into assuming that it's got a '7th' in it. It has not; it's a flattened 7th..! It's only through historical usage that the chords have these names, not 'logic', so get used to it..!
As for extending further these chords, we are simply adding another alternate note from the diatonic scale, so, for the 'C' scale, adding a 'D' above the octave gives us a 9th', adding an 'F' is an 11th, then adding the 'A' is a 13th. No point adding further, as we'd only be repeating the notes we started with, but two octaves up. With these 'extended' chords, many other nots would be left out (usually the 5th and 7th...), as too many notes spoil the broth become muddy.
I wouldn't worry about 'modes' for a while; they are not really 'mainstream', so get the basics down 'pat' before following down that rabbit hole. Interesting as it is, there's much more practical stuff to assimilate without delving further into It.
Hope this helps.
Douglas