Launching the game results in a black window and a crash, see this log file. Not sure if this is the "AVX" instructions faults or something else. FMOD seems to load at least.

Here is the backtrace:
=>0 0x00000001401fa412 in flower (+0x1fa412) (0x000000000021c5f0)
1 0x00000001402139d0 in flower (+0x2139cf) (0x0000000140390ce8)
2 0x000000014006bb0f in flower (+0x6bb0e) (0x000000000021c730)
3 0x00000001400a1863 in flower (+0xa1862) (0x0000000000000000)
4 0x00000001402ab247 in flower (+0x2ab246) (0x0000000000000000)
5 0x00000001402ac746 in flower (+0x2ac745) (0x0000000000000000)
6 0x00000001402ad8ca in flower (+0x2ad8c9) (0x0000000000000000)
7 0x00000001402a9126 in flower (+0x2a9125) (0x000000000022ffd0)
8 0x00000001402bb0b0 in flower (+0x2bb0af) (0x000000000022ffd0)
9 0x0000000140308146 in flower (+0x308145) (0x000000000022ffd0)
10 0x000000007b4824a0 PowerClearRequest+0x152f() in kernel32 (0x000000000022ffd0)
0x00000001401fa412: movq 0x0000000000000050(%rdx),%rax

I tried on another PC and it is working there (AMD Ryzen + AMD gpu) with Proton (Steam Play). The game doesnt respond to controller input though :(

That drove me nuts as well.

I have a usual logitec gamepad, the game wouldn't recognize. So I usually grab xinputemu, to let the game think, its a 360 pad what worked fine in the past.
Go to https://www.x360ce.com/. After you configurated your gamepad, just put the required files into Flower's folder.
Now comes the tricky part, that was kicking my ass. By default, my dll was named xinput1_3.dll - but the game doesn't work with that.
You can also name it:
xinput1_4.dll
xinput1_2.dll
xinput1_1.dll
xinput9_1_0.dll
So I just copied the dll and renamed it, until I had all of them at once in my folder, so I can't say, which one is the right one for Flower, but anyway, the game will accept your gamepad and lets you play like you wish to.