There are actually two separate questions IMHO:
1. Do you need to present the actual membership card?
2. Do you need the info on the membership card?
If it's only 2, then I guess you could scan the front and back and put it somewhere where you can always access it (even if there is no cellphone connection and no internet connection, or they're a pain to use, at that moment). Or make a dual-sided photocopy and keep that in your wallet instead, to save space.
But even though I rarely need to present them, I find it easier to keep them rather than just scan them. Very rarely I may need to present one somewhere, but it's too hard trying to forecast which one when. So it seems simplest just to keep them.
Now, whether to keep the all on you at all times, or keep some on you and others at home, that's a separate question. Certainly if you're traveling somewhere where airline X doesn't fly and where there's no program Y hotels, there's probably no need to take airline X's card or hotel program Y's card with you there.
Btw, since I have credit cards from so many of these hotel programs, I have the same per-trip "pruning" process to figure out: Which hotel credit cards do I need, and which don't I need, on this trip?