It's my understanding that if you don't pay for your insurance policy, the policy gets cancelled...They don't come after you.
Based on my personal experience and my understanding that most companies operate about the same, the insurance would cancel on a certain date and usually after a short grace period, of around 15 days where the insurance company would provide coverage for a loss but the customer would have to pay the policy current before the claim would be paid.
While the OP never said the amount, he should have only been dunned for around 15 days of coverage during that grace period where the insurance company was technically providing coverage, but for sure not the accelerated amount of the full policy. There might have been a policy termination fee that would have been per the contract, but even that would not be much.
When the OP got new coverage and he posted there was no gap in coverage that led me to believe that new coverage was started with another company on the date his other policy expired or stopped extending coverage.
The no gap in coverage would cause his former carrier to probably waive the 15 days or so because the customer was covered under another policy and if there was a loss the other company would be primary and therefore technically the former company was not providing coverage.
As stated, the OP never confirmed this but the post is written like this is what happened.
But you are correct, the insurance company will just cancel the policy on a certain date and might give a small grace period and then should not come after the full policy amount.
Of course it could happen, but I've never seen a company go after the full policy as they would have to answer to the dept of insurance. Now their CA could, as many do as we know, fudge the numbers.
Maybe the OP can confirm that he was only getting dunned for a very small amount (renters ins is dirt cheap) and for only a very short time period.
The dept of insurance power and authority generally keeps an insurance company from trying to sneak in a few extra hundred bucks. Charging a customer when a claim would have been denied due to non payment of premium would be an epic beat down and fines from the dept of insurance (at least in my state and from my experience).