The insurance is with anyone. They own the house, not you, and so they want to ensure it's not going to burn down (or more likely get washed away in a flood, where I live) and be irrecoverable, so they require you have the home insured. They care naught for contents insurance, just the house/building.
But if they're the ones that want the building insured, it seems like it would be better for everyone if they're the ones that source the insurance.
Is it different in the US to the UK? Surely you own the house and have a liability on the mortgage?