The US is extremely far behind in banking. It's a whole other world. Also there seems to be a culture/tradition of paying bills by cheque.
The UK enjoys free banking for people - free transfers, direct debits, automated payments, etc. Transfers show up in a couple of hours to most other bank accounts. Even businesses get most of that, including free banking for a year or 2.
Most 'bills' - water/gas/electric/cable/mortgages etc are all setup and automated electronic transfers - direct debits. (free of course)
Giving out your bank details to allow someone to pay you is fine and secure.
The need for cheques in the UK is incredibly small now.
Also, sure - people will probably still use cheques "for show". On things like charity TV programs we'll still see the big oversized cheques showing how much they raised. But it's just for show. The money is all sent electronically.