I don't think pet insurance really can be compared with other types of insurance such as car/house etc.
When a car gets pranged, it goes to a garage, the garage makes an assessment and contacts the insurance company (during office hours). The insurance can say "yes, go ahead" or "wait until we send an assessor". They can even just say "do nothing, we'll write the car off"
With pets, the treatment is often a life-saving emergency (or sometimes something else is discovered during surgery which needs immediate attention) and can't wait for insurers to give the go-ahead/wait for one of their assessors. The treatment is needed immediately. If the vet does the treatment and then the insurer later decides the claim is invalid or the cost exceeds the policy limit then they're stuck having given expensive treatment with no money to pay for the costs.
As other pp have said, some insurance companies are better than others and therefore vets can be prepared to deal direct with those ones. I see that as a goodwill gesture. It doesn't surprise me that some vets want the payment first and let the owner claim it back from their insurance.
Likewise with travel insurance, if you have a medical emergency whilst abroad, you are usually expected to pay up front (or at least a good part of it) and then claim back. Why should an A&E department in another county faff around trying to claim money back from a UK insurer? They want the money before you leave the country!
The other thing with car/house insurance is that they also have a list of approved repairers. Use one of their approved repairers and the process is much quicker (with the insurance paying direct to the repairer). Go to a garage not on their list and the claim/repair will take longer. Is there such a list for pet insurers/vets? I think the vets have to create their own lists based on bad experience (and good too).
The lesson I've learned as a pet owner is before getting a pet, make sure you have enough money (or credit limit) to cover an emergency bill. That way, whether the insurance will cover it or not, my pet can get the treatment they need immediately and the vet can be paid for providing it.