Tow truck drivers DO NOT have to take your car where they tell you to. It’s your car and you have the right to choose. Talk to the tow truck driver or police officer and let them know where you want it taken.
If you have the vehicle towed to your house, you may have to pay for the tow fee upfront but that beats having to pay daily storage fees + administrative fees at a storage lot and your insurance company should reimburse you.