I don't doubt the system checks for an MOT (if required), but I've had a couple of instances recently that make me pretty sure it no longer checks for valid insurance.
An irritating anomaly is that you can retax your car automatically each year, but only if the rate isn't zero ... because there is no way of setting up a direct debit for £0! Until this year, we had four family cars of which three were zero tax. My old Mazda2 at £20 per year would quietly retax annually with no intervention, but I would have to go online for each of the other 3 to "buy" the free tax. I have filled in the feedback form about this more than once but never had a response.