It is clearly possible to design a crypto-currency that can scale way beyond VISA/MC/PayPal and remain decentralized for mining. Don't expect it from Bitcoin though.
But that is not what will prevent offchain from proliferating, although it is a first requirement. Offchain is used because it is more attractive, convenient, and easy for the user.
I believe there is a solution. But it is currently one of my secrets that I am not yet willing to share publicly.
eMunie scales pretty well (did a million transactions within 24 hours in a beta test some weeks ago). They also have anonymity, though I cannot comment on how that is implemented. Would be interested to know what your impression is.