Falcon 9 wasn't designed for immediate reuse. Immediate reuse is just a theory right now, but SpaceX has an excellent track record of turning their theories into reality.
SpaceX did purchase some oil rigs with the intention to turn them into launch platforms, but later abandoned the idea. It's probably something they will return to later once Starship is flying regularly. You're right that avoiding the boostback burn is a big advantage. But maybe they don't need to bring the booster back after it lands on a platform, it can just launch again from there. Maybe they could have a bucket brigade of launch platforms ringing the Earth!
The issue is logistics, and in this case isnt an easy solve.
You have to get fuel, and the rocket, out to a pad in the ocean, and have to deal with a rocket lift on varying conditions.
If you dont want to do most of that, then the only option is putting your manufacturing on the rig too, which negates lifting a rocket, but instead makes the rig huge, and requires having a train of ships in and out 24/7 to keep it supplied.
There was even a company in the late 90s that tried oil rig launch platforms and ultimately abandoned it.
Immediate reuse means significantly less than 24 hours. Falcon 9's current cadence is fast enough to meet their current launch demand and doesn't need to improve, especially with Starship on the horizon. On the other hand, Starship will need to launch repeatedly in a short amount of time for orbital refueling to work.
So it was declared as a target, was possible but wasn't done because no demand? Knowing SpaceX they'd do it just for bragging.
The fact that it's required for Artemis to work, and the amount (nobody knows exactly but lower bound is like 15) of Starships required to launch in quick succession just highlights how risky, to the point of unsoundness, the project is.
They don’t need rapid reuse for Artemis. It would certainly help but Starship can just hang out in orbit for a few weeks while they do what they need to do to launch all their rockets.
It really depends - you can do direct injection to GEO now with cryogenic stages. And IIRC Soviets did some tests with kerolox stage (that should eventually launch on the ill fated N1) around the Moon, meaning multi-day flight times with liquid oxygen on board.
If you do the thermal design right, possibly use a sun shade (space is a large thermos bottle after all) or even use active cooling to remove the little heat that gets through, then it should work just fine. :)
The heat shield probably provides some degree of insulation from the solar radiation as well, and it's only needed on one side if you're far from Earth.
SpaceX did purchase some oil rigs with the intention to turn them into launch platforms, but later abandoned the idea. It's probably something they will return to later once Starship is flying regularly. You're right that avoiding the boostback burn is a big advantage. But maybe they don't need to bring the booster back after it lands on a platform, it can just launch again from there. Maybe they could have a bucket brigade of launch platforms ringing the Earth!