I’ve not had this happen in a Tesla but did have this happen in an Audi. In that case, the dealership immediately opens a claim via your insurance (rather than pursuing the other party) in order to make it more likely that they recover their money, and then they let the insurance sort out fault and subrogation.
In my case unfortunately the other party was uninsured and my insurance ended up footing the bill, and I had to pay my deductible.
But even with Tesla, the loaner paperwork that you acknowledge definitely states that your insurance becomes primary. And at least in the US, car insurance policies generally cover the owner driving another person’s (or company’s) car.