Pub near me has sign saying no cash, as does restaurant chain I like for breakfast. It's legal although I hate the fake way they say "for your protection" with a vague implication that cash is dirty. It's like you say cash is a business problem for them is the real reason.

Also I am sure that if you got them to pour drinks or bring meals then say "Oh No! Sorry I only got cash" then they would certainly take your cash.