Does the idea of a safe, regulated, core set of activities, and a less safe, less regulated, non-core make sense?

No.

The idea of a safe, regulated, core set of activities with access to the safety net (deposit insurance, central bank lending) and a less safe, MORE regulated, noncore set of activities which DO NOT UNDER ANY CIRCUMSTANCES have access to the safety net – that makes sense.

Sheila Bair in Financial regulation made simple..