How does a sitter ask for reimbursement for damages that were done to their home by a dog staying with them?

I do not agree with your belief that owners are intentionally withholding the truth. It's somewhat unpredictable to know exactly how a dog will adjust in a new environment or any place other than their own home. When you add that the new person who the dog depends on for its care leaves them, it seems logical that the dog may have some issues with that - which may result in damage if you haven't prepared a safe-guarded dog-proof environment. Many sitters utilize indoor gates, fences, playpens, and crates to achieve this.

I wouldn't leave a screen door open even while I walk one dog and one stays behind - the sounds and smells of human, dogs, squirrels outside is too hard for many dogs to resist.

It's unfortunate but Rover insurance nor any other independent insurance provider will not cover damage to your home when you are boarding a dog or dogs. On the positive side any damage can be claimed when you file your taxes at the end of the year. On the negative, many dogs don't have the best of manners when placed in a new place for boarding and the sitter must be prepared to deal with how to keep their home safe from damage.

Damage is just a part of being a dog sitter and many of us have worn your shoes.

as a self employed person, it is up to you to assume liability for damage to your home. this can be a tax write off for buisness losses. animals are unpredictable and you have to allow for that.