While it is hard to answer any specific question with one hundred percent certainty, the key to understanding is remembering who's responsibility it is.
If a driver has insurance and they cause damage then you need to remember that you can go after their insurance carrier to pay for the damages. However, the one key to remember is that even if the driver of the stolen vehicle doesn't have insurance, they are still responsible for the damage. When it comes to responsibility, the driver might even have insurance, but then again you need to remember that their auto insurance policy will probably have exclusions to protect them from any sort of theft and being involved in crimes (just like this case).
If you are wondering who will inevitably wind up paying for the damaged home, then you need to think about the likelihood of things being paid for by the end user. Assuming that the driver of the stolen vehicle's insurance company doesn't step up and pay the claim, and that the driver who stole the car doesn't have the money or financial liquidity to pay for the damage him or herself, then the insurance company that insures the home will make the payments to restore the property.
That being said, they will no doubt subrogate and seek out the return of their expenses in the form of legal action. Even if they seek out restitution payments in the amount of $10, $20, or something even less, they will want to have all of the money returned to them and they will eventually recoup everything that they pay.
To make a long story short, if the homeowner has insurance then someone will pay, and it should not be them. If someone did damage, whether or not it was with a stolen car, they will still be responsible.