Last time I looked into this in any detail (c. 2003) there were four Lloyds underwriters, plus one insurance company with a name like AIG or BAIG.
There were of course many brokers, and there still are a fair few despite Haywards (a very good company which I insure through, BTW) having absorbed some.
If there was just one underwriter or insurer, we would see skyrocketing premiums, which is clearly not the case. They have been falling recently, which indicates competition.
The large # of brokers can confuse people, who contact multiple brokers thinking they will get competitive quotes!! But sometimes one goes through 2 or more brokers in a chain, when they split the commission between them.
I think it is right for insurers to not pay out on blatently illegal pilot or blatently illegal aircraft, otherwise people would take the micky.
What would greatly worry me is if they did e.g. a back to birth maintenance audit, because there is not one plane flying, not even those landing at Heathrow right now, which would pass that without any irregularities. So there is a grey area here, which presumably they have a corporate policy to not probe into. But there is NO evidence they are doing that - ever.
There is also NO evidence they are getting out due to alleged poor airmanship. I've never heard of it. Rumours, rumours, sure, but forums are full of them. Like the SR22s busted for illegal IFR without an ADF/DME (totally false).
I think that keeping the papers legal, and paying the premiums etc, one is covered.