I am - as usual - late to the convo.
I did, once upon a time and many years ago, deal with football liability claims, amongst other things. Way back before transgender players were ever an issue..
Football Clubs usually get their cover through some sort of scheme, often at county level, to make it cheaper. It will provide PL for physical injury (not between players in course of game) and third party property damage, EL for injury to employees, and officers and professional indemnity to cover wrongful acts (like discrimination), and claims for libel and advice given. Player to player liability and accident cover are usually available as add ons.
I doubt insurers had much sway in affecting policy at the FA. I am not sure that the inclusion of transgender players presented a significantly higher risk to insurers, despite posing an increased risk to female players. As long as clubs were following the FA guidance on transgender eligibility it would have been hard for anyone to pursue a claim for an injury on the pitch, unless the player causing the injury was particularly egregious in their foul play. Just being bigger and stronger wouldn't be enough to make the player liable for injury through standard play. Just for context, I saw just one player to player claim for the scheme I dealt with and it got booted out (pun intended) because the claimant had caused his own injury. He'd gone in for a completely wild tackle, totally misjudged and busted his leg. Ref had red carded him on the stretcher.
I do think the FA are responsible, at least morally, for women who have sustained serious injuries from male transgender players in the past, although whether they could be held legally liable I have no idea.
I think insurers and FA feared claims for discrimination, both the financial effect and the publicity, much more and thus inclusion of transgender players in the women's game would have been fully accepted by insurers.
Basically fairness and safety for women didn't come into the equation.