I have visited 30+ countries in the last 11 years.
(Much of mainland Europe, UK, USA, quite lot of Asia, Australia.)
I have never been asked about my travel insurance by any official (or anyone else).
Which is not so say that your experience will be the same.
The closest I've come is when driving a 'Eurolease' car, which is fitted with telltale "I am a tourist" red number plates :-(, and entering Slovenia from Italy, the border guards / customs men said our vehicle insurance may not be valid and we had to wait while they checked elsewhere. Initially they 'put on a somewhat stern face', but once they had verified that the insurance was OK they were most friendly and cooperative.
In China I was bitten by a playful guard dog and had to have Rabies injections. They did not ask about travel insurance and I suspect that as the full charge for an initial anti-Rabies course was under $US10 they would have not been happy if I'd asked. (Next 3 required shots in following weeks and months in other countries cost $100s each.