The problem with your itinerary is that you are driving vast distances with not much to see or do along the way. For instance, Bs. As. - Bahia Blanca has very little of touristic interest. Then Bahia Blanca - Puerto Madryn again, and the worst, Puerto Madryn - El Calafate. I made the mistake of thinking it'd be great to take the bus from Rio Gallegos to Bs. As. and see a bit of the country! I figured the first bit was night, and I knew it was uninteresting, so then I'd have a whole day to see the sights. Well, it was 12 hours of seeing the typical Patagonian shrubbery grow from about ankle height to knee height. I hoped that Bs. As. province would be more exciting, but by then it was night. Luckily for me, I had another chance at that trip... and no, you don't miss anything from attempting to sleep through all of it. It's none of it unpleasant terrain (except for Commodoro Rivadavia, which is a dump), just long, straight, and monotonous.
Once you hit the south, you can enjoy your trip a lot more, and here a car will really shine. You can do things like visit the Cueva de las Manos and other hardly visited spots on the beautiful drive along the Andes from El Chalten up to Bariloche. It's long, but there are now actually vistas, and places that are worth stopping for a few hours or days. Similar for Bariloche to Mendoza. Depending on how much time you have, San Juan is pretty too, and then you can stop in Cordoba on your way back to Bs. As. (and also Rosario, although it's less interesting than Cordoba).
Even so, it is a LOT of driving for a 4 week vacation. Just to get south, on the slog mentioned with no interesting sights, I'd still do the drive south in 4 days, with an extra 2 days on Peninsula Valdes. If you are 2 drivers and both used to driving long distances, you can cut a day off there by doing the second half (from Puerto Madryn to Calafate) in a single slog. If you're very courageous, you can do the first half in a single drive as well, but I would suggest doing it in 2 and taking it easy (insofar as you can call driving 700km a day taking it easy): get a bit used to your car, the roads, and other drivers.
Oh, and in Patagonia beware of the winds.