What is "weight" relative to a civilisation with advanced gravity manipulation? Starships have been proven to go from FTL to a full stop in less then a second or two, with that level of cancelling out inertia why can't a Galaxy-class dance around across six degrees of freedom?

On top of that, the most restrictive factor in manoeuvring is how much G-force a pilot or crew can take. In Star Trek, that G-force is always zero.

The Doylist answer is simply that when Trek started an audience couldn't have been expected to understand 6DOF movement and so the ships handle like aircraft and are governed by the same restrictions. We're conditioned to think of larger starships as lumbering beasts akin to the old battleships of yore but there's no in-universe reason for it.