I must say though "right on its launch tower" is fun and all...
but things would have to get pretty extreme in the "routine" dimension for that to be very useful. If there are 20 first stages and 5 land/launch towers, for a first stage that only spends an hour in flight in between inspections.... well are you going to keep them parked on a scarce launch tower for maintenance? The towers with fill/launch infrastructure (such as reinforced concrete, fuel tanks, cold filling system, deluge water system) become the bottleneck. It's more likely then that the 1st stage lands, is safed, then is taken a couple hundred yards away for inspection and maintenance while the next in line is moved to be stacked for the next launch. The inspection / maintenance would have to be truly minimal (think airliner) to keep it right there on its own relaunch tower.
"No landing gear" is more key, compared to Falcon 9 - because of the effort toward minimal launch cost.
but things would have to get pretty extreme in the "routine" dimension for that to be very useful. If there are 20 first stages and 5 land/launch towers, for a first stage that only spends an hour in flight in between inspections.... well are you going to keep them parked on a scarce launch tower for maintenance? The towers with fill/launch infrastructure (such as reinforced concrete, fuel tanks, cold filling system, deluge water system) become the bottleneck. It's more likely then that the 1st stage lands, is safed, then is taken a couple hundred yards away for inspection and maintenance while the next in line is moved to be stacked for the next launch. The inspection / maintenance would have to be truly minimal (think airliner) to keep it right there on its own relaunch tower.
"No landing gear" is more key, compared to Falcon 9 - because of the effort toward minimal launch cost.