If you want to visit Torres Del Paine (which I strongly recommend) then primary access is via Puerto Natales not Puerto Arenas (but the nearest airport is Puerto Arenas) and I’d recommend getting to the National Park by bus rather than car as they are frequent and it is the NP itself that is special rather than the drive there.
But the key thing is where you are starting from? There is southern Chile and then there is really southern Chile! Transport can take a long time in those parts, flights and coaches are not necessarily daily and this is a busy time of year so planning ahead can be important (especially if you want to stay inside the NP).
We didn’t go to Puerto Arenas but did stay several nights in Puerto Natales it is likeable and atmospheric frontier town (go to Aldea Restaurant for food) but TdP is certainly a bigger reason to be there. There are several places you can hire a car at short notice in town.
But all of this needs quite a bit of time. If you are time limited then the Lake District or Santiago are likely to be better bets. There is a good mtb guiding company operating out of Santiago if that is of interest.
Absolutely no need for a pick-up, you can drive to and from the park but not really through it. Dirt roads are all in reasonable condition.
You’ve posted since I did – If you have the time do the W trek. Needs planning and booking and it is very busy but it was one of the highlights of our four month trip. Driving around the park does not compare (we did that the following day).