I actually didn't check the app or the browser store.....Didn't even think to do so. I WILL TRY TO REMEMBER for the next time, because I have every reason to expect this to happen again. Regardless, it's BS that it's happening anywhere. I used to claim on mobile often, because it would pop in my head to do so late in the month when I was, say, shopping for new socks. I've really gotten in the habit of using the store on the console the last few years though.
Screw me, you're right! As promised, I claimed this month's PS+ games on PS4 and while it was possible to simply add GreedFall to library, I had to start a download in order to claim Shadow of the Tomb Raider. I wonder if it comes down to specific publishers being assholes (Square Enix in this case and Activision in June and August) but no matter what the reason is, I agree with you that it's dumb. Use the store in the app or browser (or on PS5 when/if you get that) instead as you'll always just be able to add to library there.
Shadow of the Tomb Raider on PS4 (can only be claimed by downloading).
GreedFall on PS4 (can be claimed without downloading – "Tilføj til bibliotek" means add to library).
Maneater (and all other games so far) can be claimed without downloading in the store in browser and app.
Regarding the issue where you have to write your password each time you want to claim a game, you just need to disable that feature.