Most tow truck drivers are owner-operators (or are hired drivers for an owner-operator), at least around here.
So I don't expect the Tow Company to step up to the plate. However, it's a great opportunity (and I'd be shocked if a lawyer would not point this out) to have the Tow Company's name on the door because it's always best to name as many targets as possible in the lawsuit, because they are (or at least might be found to be) "jointly and severally" responsible for any judgement.
Basically, that means you want someone with deep pockets or assets that can be seized because any one defendant can be made to pay the whole judgement, rather than just one poor-ish person who can reduce his assets below the minimal living threshold and therefore never, ever, actually pay out the judgement. The old "blood from stone" defence, which is not an actual defence so much as a strategy, but none the less a real problem.
So, it's a nice stick to have. The Tow Company's lawyers will no doubt say the same thing, making them leverage as they can be counted on to pressure the owner-operator to settle early, to the car owner's satisfaction and benefit.