I think I may have a solution that addresses both of those, and which reinforces the theme as a bonus: an imperfect view of your deck's statistics as a report of the various "silos" of your deck.
I have a prototype working, and it looks like this:
When you press-and-hold on a deck (either your discard or your draw), it pops up this dialogue and animates the histogram into place. Each card in the game is assigned to one of these eight "silos", such as industry, religion, military, etc., and the histogram shows the relative prevalence of cards in each silo in that deck.
This way, you can't know for sure how many you have of each card, but you can get a pretty good idea of what's coming up and what you've already gone through. It should also really help people survive the early game if they're agonizing over whether to buy that extra priest or something.
I'm still not 100% sure I want to include this. I want to play with it a good amount - and get everyone's feedback on it - before I commit. But it should show up in the next build for everyone to try out.
Thanks for the thoughtful feedback, everyone! You're really knocking it out of the park on some of your ideas.