Additionally, have you seen the hatch on the Lunar Starship? That thing is ~30m above ground and only has a small platform. Loading and unloading that thing would be slow and dangerous compared to simply dropping cargo from a flying aircraft that doesn't even need to touch down or just driving on and off the ramp in case of landed planes.
The concept of rocket transport isn't new either - it has been considered several times and always turned out to be a bad idea.
To see where some of this comes from, look here: [0]
Well it needs fueling infrastructure but if the mission is designed sich that there is enough fuel left to fly to some air craft carrier protected barge nearby the rocket might not be stuck there
Land near combat and take off to America or Europe for surgery.
The flaps & actuators might be worth taking, but the top ones are inconveniently high up unless you tip it over. Probably the electronics is worth shipping home if it's easy enough to strip off.
I don't know why the thrust plate isn't titanium. A 30 ft diameter thrust plate would in any case be inconvenient to ship home, from most places.
The logistics of such system would be hell, too - Concorde only had limited routes due to noise. A rocket of Starship's size is another thing entirely. You'd basically have to launch and land it offshore so the logistics getting to and from the launch site and into the vehicle would kill all benefits.
Not to mention the simple things like weather, closing airspace, and the lack of cross-range capabilities, which would make emergency landings or evading bad conditions on the landing site pretty difficult.