With Chase quickpay, only one member, either the payer or the payee, needs a chase account.
I use TD bill pay to pay everyone and never had an issue. In the account number field I just put 0 and it's fine. It's anoying because, for example, I once sent someone a check and of course the money leaves your account right away. 4 months later he tells me how found the check in his drawer and TD never even let me know that the check was never cashed, and there's no way to tell.
I actually prefer if there's someone I pay often, I just ask for his acc#, routing #, and checking or savings, and ACH the money to him. That was it's there quickly and I know he got it.