Not really. As far as I understand, the booster is only for launching from Earth. From elsewhere (moon, Mars) Starship can launch alone.

Then maybe it would also require that much less thrust?

I thought the issue with throwing up a bunch of dust is that the planet this rocket was starting from is inhabited.

Not quite. It is speculated that the rocket engines were damaged by chunks of the pad hitting them. That is a danger to the launch itself even if the dust wouldn’t bother anyone else.

A flame trench would probably be a lot more inconvenient to them. Right now they can freely move around most of the launch pad. This is probably important for their rapid iteration, which is what is going to get them to orbit as quickly as they can. If a big chunk of the launch pad is consumed by a trench, that area becomes inaccessible by their platforms, cranes, etc.

Plenty of types of cranes can span a trench.

Isn't the water table too high at that location? If that's true then it was never a realistic possibility to build a trench.

The main problem isn't Starship with its six measly engines on a lower-gravity celestial body (e.g. faster lift-off).

The main problem is the Super Heavy booster, tearing the ground apart with 33 engines at full thrust while struggling against 1 g downwards. This is a very Earth-specific phenomenon, because of our relatively high gravity. You can easily do a Single Stage to Orbit launch with just Starship on Moon, Mars, Europa etc. It is very likely that Super Heavy boosters will never leave the Earth.

The upper stage, with 6 engines, is the part expected to land on the Moon and Mars and take off. Still, that's going to be challenging: Lunar regolith wants to clog and grind everything it gets into.