Slightly off topic but I am the only one who finds it a reflection of government priorities that it sends me a reminder to pay my car tax for free, but wont send me a reminder to have the road worthiness of my car checked without charging me a fee.
Nope, we now have an entire nation that is totally devoid of any sense of responsibility, it isn’t just you. ;)
No MOT does not invalidate insurance.
Your car can be roadworthy or unroadworthy with or without an MOT, so why would it?
At worst, in case of total loss an insurer may reduce the payout as a car with no MOT is less valuable than one with.