In theory I can do this with standing orders / direct debit in the UK and there are some subscriptions where "cancelling the direct debit" is the official way to cancel. There should be no need for firms to reinvent recurring payments and store card details for their own ad hoc system. I don't know if it might disadvantage some people not familiar with managing direct debits though.
However, many years ago, after an hour on hold failing to cancel Virgin ADSL I just cancelled the direct debit instead. They put a debt recovery firm on me! The direct debit was charged at the start of each billing period so it wasn't a non payment thing. I recall there used to be more indefensible "notice periods" for cancellation which were just pure scummy ways to force feed unwanted services but I don't think this had one.