I went about 18 years ago -- we traveled independently and it was amazing.
We bought a "ticket pack" on LAN which may not be available any longer, but really helped with keeping the air travel costs down. We met up in Buenos Aires and flew to Bariloche. Did dayhiking in Bariloche -- we took the ski area gondola up and then hiked down -- and then took the Cruce de los Lagos to Puerto Montt. After renting a car and spending a few days in Chiloe, we flew to Punta Arenas and then bused to Puerto Natales. We stocked up on food and other camping supplies (e.g., fuel for camping stoves) and then took a bus to Torres del Paine. We hiked the "W" in Torres del Paine -- we camped, but you can also stay in the refugios. It seemed like most of the people in the refugios were with a trip operator, but at least at that time, you could also reserve space in a refugio as an independent traveler. Upon our return to Puerto Natales, we took another bus to El Calafate for glacier trips -- we did the boat trip on Lago Argentina to Uppsala glacier but you can also go to Perito Moreno from there -- and then on to El Chalten to backpack into the campground on Laguna Torre (I think) -- with an amazing view of Cerro Torre.
After this, we flew to Calafate to visit friends, and then on to Santiago to go home.
We spent a lot of time on buses and were in and out of Puerto Natales -- but we had a month for this trip. We also ran into a tour group on multiple occasions that was doing much the same itinerary and taking the same bus routes, for whatever that is worth. We went in December and weather might be best described as variable. Some unbelievably sunny days, others socked in with clouds and fog, others extremely windy. We were prepared with waterproofing and good camping gear.
In my experience, this trip was completely feasible as an independent traveler, but if you want to do glacier hiking or more involved backpacking, a tour operator might be helpful.