I think all of Ben’s and my proposals have assumed (without saying explicitly) that you shuffle within each suit. If you do that, then I think your concerns all go away? Let me know if you don’t think so.
I think Ben’s proposal is: between rounds, it takes a while to split the whole deck into suits, all hearts in one pile and all spades in another and so on. Instead you can just pick out four hearts, and four spades, and so on, and remove 0/2/2/4 cards from those piles, and shuffle the rest back into the deck. But no matter how you shuffle, I don’t think you can do that without leaking information.
I think all of Ben’s and my proposals have assumed (without saying explicitly) that you shuffle within each suit. If you do that, then I think your concerns all go away? Let me know if you don’t think so.
I think Ben’s proposal is: between rounds, it takes a while to split the whole deck into suits, all hearts in one pile and all spades in another and so on. Instead you can just pick out four hearts, and four spades, and so on, and remove 0/2/2/4 cards from those piles, and shuffle the rest back into the deck. But no matter how you shuffle, I don’t think you can do that without leaking information.