Is this a sensible way of using an IoC container?
No. What you are doing is called The Service Locator Antipattern. And it's a bad thing.

The basic rule of thumb: Don't inject the container!.

The IoC container is supposed to be some sort of overseer, wiring your objects together at the bootstrap and configuration level. It isn't an object that's supposed to be injected into other objects, to create objects there.