I'd like to replicate this for myself, but since it can take me a long while to find out which games were written in Flash and which were not, perhaps things go faster if you post a few links to games you tried.
What I do know about Flash is that the entire concept has been deprecated, and that causes trouble on more and more sites now. Still I suppose downloading an .swf file should still work... Right?
Okay, I don't see a download link at all, as these games are (as I suspected) supposed to be played inside a browser.
What I see is "Adobe Flash is required to run this project in the browser. Enable Flash".
Now since Flash has been deprecated I never bothered to even try to install it at all, and it's not even fruitful to get this to work, since Flash will finally die in 2020. I don't know if itch.io has any plans to make the .swf files downloadable (doing so could violate copyrights of the game's developers, so they gotta be careful on that one), but given the way these games have been uploaded here, and that Flash is going to die, that these games will die with it. The Microsoft browsers (IE and Edge) already stopped Flash support or will do so this year.
I'm very very sorry... But if this a fault on your computer I don't think you'll find much support, and if this is a bug in itch.io, I don't think it's in the site's interest anymore to get it fixed.
In side the client app for itch.io I see the same as I see in the browser, and everything looks like it was never supposed to be able to download flash games (which is actually also what I suspected). Perhaps you came past a few games which were specifically configured to be downloaded before... I've checked this even out with a downloadable game, where the download/install button does appear.
It looks to me the app was never intended to grand the possibility you claim it should have (it really doesn't look like that to me, at least). Oh yeah, the app also wants me to install Flash....
