[...] This would be solved by putting a visible blockchain counter in each gaming room though, or maybe writing it into the chat.
It's a tradeoff between covering absolutely every way the house could cheat, and not flooding the chat with too much information.
Instead of "the most recent block's hash" they could just use "the next block hash after the block containing the hash of the pregenerated shuffles". That's both unpredictable and fixed.