The exact way the Romulans use their artificial quantum singularities is not explained to us - but "Face of the Enemy" specifies it as a power source rather than a drive system. Then again, the knowledgeable double-defector in the episode also says that if the power source hiccups, one may observe a "magnetic" disturbance - probably the same sort of magnetic as in "magnetic boots", that is, an attractive force that isn't necessarily electromagnetic in nature. Might be the AQS power source moves the ship by direct manipulation of "magnetism" or gravity. Might be it just produces stealthy power for conventional warp engines.
FWIW, Romulans still mine dilithium on Remus...
It's true that if a pressor/tractor beam can push or pull an object then it can push or pull your ship where the mass of said object is greater than the mass of your ship.
Or then regardless of such mass (if any). After all, A locking a tractor beam on B creates no forces on A: say, Wesley can lift a heavy-looking armchair with the strength of his wrists along a momentum arm several meters long in "The Naked Now", by locking a tractor beam on that chair. There's nothing as simple as Newtonian feedback involved there.
There are also limitations. The field generated by the emitters is narrow and not omnidirectional.
Which would be great for moving in a desired direction, I guess.
The ship had insufficient power to divert the asteroid in the Paradise Syndrome and burned out its systems in the attempt.
A rock "the size of Earth's moon" might be a tad high a goal to set. Why worry about that when you only want to move a starship?
The creature from "Obsession" used gravity fields for FTL and sublight propulsion so in some sense early Star Trek writers understood that possibility.
Good point - the heroes seem to think that using gravitational force for propulsion is worth a special mention, suggesting it's not something they themselves do all that often.
Or is it just that a "creature" doing this is exceptional? Probably not - Spock later claims that doing so goes hand in hand with the ability to penetrate deflector shields, and our heroes and their usual opponents do
not penetrate deflector shields as a matter of routine. (But perhaps their torpedoes do, say?)
Timo Saloniemi