NASA is doing everything you BUT the actual landing part of the mission. The launch, the transit, lunar orbit, SpaceX will land, NASA does the science mission, SpaceX docks then back with Orion, NASA does the return flight back to Earth, landing on Earth. NASA was ALWAYS going to get a private company to do the landing.
This is the generic plan. The HLS will be SpaceX's 'Starship'. You can ignore the stuff about 'Gateway'. Gateway for initial landings. Orion would directly dock with Starship.
I said the most of the hardware for a Mars trip will be from private companies. The only think NASA will probably build after SLS as far as human spaceflight goes are stations and scientific equipment.
A private company will no head a Mars mission until NASA figures out how to safely get humans from Earth to Mars. We don't don't know how to do yet. Keeping someone on the ISS is easy, actually dealing with deep space travel is a different ballgame. Radiation, supplies, disaster plans, the mental, physical, and emotional health of astronauts. No way NASA isn't micromanaging the process. Then you have landing on Mars, which no one can quite get right except NASA - and you can land people the way NASA landed Curiosity/Perseverance. It's going to take a lot of dry runs before a lander has a person in it.
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Who has what?