Swamp Thing said:
Or, would it work if we compose/improvise a melody first (or adapt it from a non-harmony-based genre like an Asian one) and select a progression from the "official" ones? Or, is it equally valid to generate one's own progression to suit the melody?
I've composed music and still do for fun.
I'll write something about my take on it all below...
To me, things often come in pieces (1), but sometimes a song can come as a whole pretty quickly (2).
1. Piecemeal composition
The pieces are different types; chord progressions, melody ideas, hooks, riffs and sometimes even just sounds and atmospheres (the "feel" of a sound that excites me).
I've got many such things around recorded as small recordings, ready to be incorporated into something if I am inspired to do so.
Piecemeal composition is fun, but it can take a very long time to do, and a bit frustrating when you get stuck. It is like building a puzzle, but the difference is that you as the composer are in control of what the puzzle will look like in the end.
2. All-in-one composition
What I mean by this is when a composition gets done very quickly. And with composition I mean the entire song (chord progressions, melodies, songs parts and lyrics). It could (but rarely does) happen as fast as one hour (or less) but not more than one day. This regretfully very rarely happens for me, but it has happened.
When you are in that very rare creative (and lucky, I'd say) mode and it all comes together like it is magic, it is extremely satisfying and gratifying afterwards. It is an AMAZING kick. And that's why songwriting can be a bit addictive, I'd say

.
Chord progressions - in general
Knowing your way around chord progressions is a very good and useful tool when composing music. If you compose a lot of different styles over the years, you will get familiar with various usual or more unusual chord progressions.
But chord progressions can also get a bit boring sometimes, since I often then say to myself "nah, I've done it before/I've heard it before". The tricks I try during such times can be e.g. (1) to color the chords differently, (2) play them differently, (3) insert other small, quick chords in between the usual ones or (4) doing something other drastic like using an arpeggio that (sort of) follow the chord progression (the arpeggio may make the otherwise usual chord progression sound more interesting).
Edit:
For me things often come in this order (but certainly not always):
1. Chord progression(s) and possibly hooks (this happens since I most often start by pure improvisation).
2. Melodies and riffs (and sound choice, e.g. guitar, piano, keyboards etc).
3. Song structure (e.g. verses, bridges, choruses, breaks, solos, intro, epilogue).
4. Lyrics
5. Recording (personal/private demo), which during recording or after multiple listens of it sometimes demands remakes of things in point 1-4, since you then may hear that the things do not come out/fit the way you like them too. It can be frustrating and annoying, but you just have to go and weed things out and/or rewrite some and don't be afraid to kill your darlings (see
https://en.m.wiktionary.org/wiki/kill_one's_darlings).