The thing with insurance these days is that most of the firms on comparison sites are "fronts", i.e. just sales/marketing firms trading as front ends for a very small number of "real" insurance firms working in the background who actually do the work, deal with claims, etc. There are relatively few "real" insurance firms working in the background, but lots of "fronts". The "front end" sales/marketing firms just deal with the quotes and getting the business, then the "real" background firm deal with the admin, management, claims etc. So they're not really "fobbing you off" to separate claims management firms - the firm you bought your policy from is just a marketing/sales agent acting on behalf of the firm in the background - your policy was always underwritten by the background "real" insurance firm.
To a large extent, it doens't matter which "front" you buy the policy from - they're effectively competing against each other, but many of them competing are using the same insurance firm behind the scenes, so you can end up with the same insurance firm underwriting the policy and dealing with any claims.
It's the same whether it's travel insurance, pet insurance, home insurance, car insurance, etc.
Eg Direct Line, Churchill, Privilege and Darwin are all effectively the same background insurance firm, being UK Insurance Limited who don't actually trade directly under their own name.
I had to claim against a third party insurance firm when a neighbour wrote off my car. It was quite funny during the claims process when I had to make numerous phone calls to the claims handlers and they had to keep asking which insurance company it was as they were acting for lots of different "fronts". Once, one of the claims handler (on their direct line) picked up her phone and answered it in the name of a different insurance company by accident as my claim was with a different one!
Back to travel insurance, I once had to make two claims against "get going" travel insurance policies due to a serious health diagnosis. I expected the worse with them being "cheap and cheerful" but the claims handling firm was incredibly good - no quibbles, I just filled in the claims forms and they quickly paid out - both holidays were expensive so I expected arguments, but no problem at all - two payments of a few thousand each straight into my bank account within a few weeks of making the claim. Before I submitted the claims, I'd phoned them a couple of times to tell them of the diagnosis and cancellation and they were really efficient. So "cheap" doesn't always have to mean poor service. I'd happily use them again.
Travel Insurance Facilities Plc is an insurance firm used by a number of "front end" sales/marketing insurance firms such as Flexicover, Lighthouse Travel, Alpha Travel, Holidaysafe, etc.