Neither does cash. Bitcoin behaves rather like cash in this respect. It has advantages over cash for the vendor: it's easier for them to count; they don't have to physically take it to the bank; it's harder for their staff to steal it. (The main reason supermarkets offer cash-back is as a way to get rid of the cash and replace it with something digital, because cash is such a pain for them to handle.)

The main drawback here is that if someone hands you a £10 note, and then changes their mind, you can give them the £10 note back. With bitcoin, you can't do that: you can't send them the output of the transaction they just sent you, because it won't be in the block-chain with enough confirmations. You'll have to give them the output from an older transaction instead. You'll need to keep a pool of old transaction outputs ready, to handle change. Much as you need to keep a pool of notes and coins to provide physical change when customers pay with cash. (Again, it's easier with bitcoin because it's digital; it doesn't take up space and all your tills can share the same pool.)

I'm not saying payment processors won't happen, just that they aren't needed.