Really almost everything I tried works. The way I proceed is the following :
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try finding the app in F-Droid. I consider this the most secure way, apks recompiled from open-source with severe criteria (no closed elements, detailed explanation on ‘who calls home’, etc.)
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when not found on F-Droid, try /e/ own app repository (‘Apps’, installed by default). /e/'s Apps adds many non-open-source apps that are guaranteed to just work, again with a detailed assessment of their confidentiality
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when none of the above works, which concerns things very specific, or national-only, install the Aurora app store (available in /e/ own ‘Apps’), configure it (basically one single setting : choose to stay anonymous or give away your google account) and get the app -Aurora just accesses google’s own store, so everything is there. Among other, all closed-source apps calling home freely -but that’s your choice then.
Given the way /e/ simulates Google’s hooks and tricks, there indeed may be some remaining apps that once installed won’t work. But honestly, with some 100 apps installed on an /e/ Fairphone 3 here I found just no failure.
The only trouble I had, initially, was trying to send myself apks from the previous phone : in that case a couple of them didn’t work properly. But once I replaced them with Aurora-installed versions, everything went OK.
Incidentally, I don’t know where to post a list of ‘verified’ apps.
There is the thread specific to banking apps that @AnotherElk mentioned above, with an editable table that I did update, but no general place apparently, or I don’t know it.
[edit] a thread dedicated to Aurora, with much more detail than that I give here : [HOWTO] Download every app in the Google Play Store thanks to Aurora Store - #3 by PoorPocketsMcNewHold