You are forgetting something...
What is the biggest thing one has to boost toward Mars, to go to Mars?
It’s not the ship — at least not exactly.
It is reaction mass and / or fuel (tho’ “fuel” mass can be minimized to some extent, with a nuclear powered engine.)
One is a LOT better off with one’s reaction mass / fuel on the Moon than on Earth, because Luna’s gravity well (to use Heinlein-like wording) is small, while Earth’s is MUCH larger. Luna has “energy of position”. If it also has extractable ice, we are in business (although that is not the only possibility.)
The Moon has something else, too: Almost unlimited solar power, unblocked by any significant atmosphere. True, in most locations (assuming no big orbiting mirrors) it is only “on” 50% of the time, but, compare to most terrestrial locations...
So... build a large magnetic catapult on the Moon. Build as many ship components as possible on the Moon, and catapult them into Lunar orbit for assembly. (A large magnetic brake and catapult could also stationed in Lunar Orbit. The surface catapult can also be used as a magnetic “brake” for incoming items needed on the Moon. This is a bit hairy, but probably much less so than landing a plane on an aircraft carrier!)
At some point, begin catapulting reaction mass / fuel, food, etc., up to the Mars Mission ship. (We’ll assume the ship is only partially self-sustaining when it comes to food and supplies.) Begin the mission, and keep catapulting supplies and especially reaction mass to the ship until it is out of range. (This could be a 3 stage process utilizing both the surface and orbital catapults.) Reaction mass, etc., could even be sent to Mars orbit, for the return voyage.
The above concepts could be scaled down if all one wants to do is send 2-3 astronauts to Mars once or twice. But where such a plan really shines is if one wants to REALLY explore Mars. It also gives the builder one heck of a weapon. The USA’s terrestrial military budget could easily be cut in half.
“It also gives the builder one heck of a weapon.”
The Chinese are working on it ...