Certainly, there should be a competition, but I would not go into banking kind of infrastructure, because then when one of them runs away with money, the entire system collapses. Just like the one we are now seeing collapsing

So I rather see it like some common "bitcoin paypal API", that would allow a merchant to hook to a new instant payment provider with no much effort.

So:
* Being a merchant, to accept payment from a new "bitcoin paypal provider", you just need to create a merchant account there and add it to your billing system (POS).
* While being a payer, you need to create a customer account at a "bitcoin paypal provider" of your choice and deposit some bitcoins there.

Mind that each of these providers would give you a different kind of security/anonymity/fees/etc.
For instance: some of them can even credit your payments, though definitely not those which did not check your ID.
And that is great - because you have a choice.
That is pure market and that is how it should work.

Of course the payment providers/processors/insurers (aka "bitcoin paypals") can come out with a system allowing off-chain transactions between them.
But that would not be a requirement and such systems would rather be transparent for the users, and the merchants.
As long as the balance on our account is correct, we don't really care about how the corporations settle their balances between each other; on- or off-chain.