I don't suggest we "split up" Bitcoin. My question was, "what prevents the rest of the world from adopting Bitcoin, the distributed ledger, but (mostly) ignoring Bitcoin, the currency"?
Nothing prevents it. And I hope we see greater use of the blockchain as a global trustless ledger moving forward (e.g., colored coins, oracle-based sidechains, etc.).
But it's important to remember that the only asset that can exist on that ledger free from counter-party risk are bitcoins. In times of crisis in this blockchain-based future you're imagining, market participants will tend to trade out of assets with counterparty risk, as well as out of oracle-based sidechains, in favor of the one asset that is itself its own backing: bitcoin. I image bitcoin then as an
attractor within a complex dynamical system; there's a constant flow of capital in and out of various blockchain assets, but these just circle around bitcoin. Bitcoin naturally becomes the reference point
1 from which to measure the value of everything else. In other words, bitcoin becomes money.
1And the value of bitcoin is tethered to the cost of real resources due to the great physical efforts (mining) required to produce them.