Same in the UK, and similarly they often cease at 11:59PM, which has always struck me as pretty unsatisfactory.

What if a plane crashes on my house, between policies, during those two uninsured minutes?

Sure, it's unlikely, but so are all the other things that might cause me to claim on my buildings insurance policy.

So make insurance policies overlap by 23 hours and 58 minutes? And leave it to the insurance companies to figure out which one gets to pay out if bad things happen during the overlap.

Most policies I've seen have a clause that says that if you are double insured for the same event then they won't pay out - which in effect could leave you with neither company paying anything. I have no idea how a court would look at such a clause, but it definitely exists.