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....