Probably not. The tower is used for horizontal transport and erection to vertical, for loading power and propellant to all stages of the rocket up until a few m

Probably not. The tower is used for horizontal transport and erection to vertical, for loading power and propellant to all stages of the rocket up until a few minutes before ignition, and for protection against strong winds. And, perhaps, loading last minute cargo (like people). So you kind of have to have it in the current configuration. And since you have to use a prepared range and pad, it's not that much additional infrastructure. One could perhaps design such a rocket, with storable propellants (liquid oxygen boils off, which is why they keep supplying it for as long as possible) and ground-level power connections. And on Mars the wind isn't really a problem. But that's a different rocket.