Should I use the company recommended by the insurance company to do my water restoration or remediation?

My answer would be the same as would you trust your insurance adjuster to act in your best interest and pay you fairly? It is always your decision on who to hire as a contractor unless your policy dictates otherwise. I would advise you to call a few contractors who do not work with insurance companies and let them explain why they will pay more attention to detail, document all of the damages, and make sure the job is done right. Remember they are not contracted regularly by the insurance carriers and are not well in most people’s opinion trying to keep the costs contained for the insurance company to help them minimize their losses. This is an opinion. My advice is to meet with a few private contractors and the carriers and then make your own informed decision.