I don’t think this is gamey at all really as you’re not getting anything of benefit out of it (gamey usually means when doing something meaningless/grindy/easy gives you small bonuses) and adding more ‘objectives’, even if with little actual gameplay pay off, would be a great way to add more interest/challenge for the player.
One game (I can’t remember what it was) gave you a different starting screen if you collected all of the collectables/whatever it was, which I thought was a good way of having something to show for your effort without it affecting the actual game play
Cards sound like a good idea, but I’d have it at a moderately completable number, something like 50, and weighting them to labs/military/subways/other harder to reach places.