Coinbase et al. may attempt to keep their users inside a walled garden that is only allowed to transact with other people in the same walled garden, but they won't get away with it in the long term.
I'm not so sure. That's certainly now how things are going now.
They are going to make the attempt. Whether because they want to, or because they are helpless pawns of regulatory forces beyond their control, is irrelevant.
In the end, it won't work. Unencumbered bitcoins will always have more purchasing power than restricted bitcoins. The price incentives this situation creates will ensure the eventual failure of their walled garden attempts.