SpaceX captures a launch vehicle for the first time through mechanical capture arms

They will never be promoted to Mars. It is only needed on the ground. Of course, because of depression.

Rocket capacity is often expressed in terms of delta-V. That’s how the rocket can change its own speed. You can think of it as the space travel equivalent of a car range.

To get from earth to low Earth orbit, you need about 9.5km/s of delta-V. To then go from this low orbit to Mars you need another 3.6km/s. Once you reach Mars you can use the atmosphere to slow down until you reach the surface. You only need 5.9km/s to fly back to Earth from the Martian surface.

Starship has a delta-V of 6 to 7 km/s. So a Starship can fly from Mars back to Earth (With a full tank of fuel), but not close enough to get into low Earth orbit.

So the mission plan is that you use the big lift to give the Starship an initial push of 2 to 3 km/s. The Starship then uses its own fuel to enter low orbit. Then you send up a bunch of tankers to refuel them until the starship reaches 3.6km/s and can fly to Mars. And then you need a gas station on Mars to fill up the starship for the return trip.

So you only need the booster for the first part, and the booster never leaves the Earth.