My quick test said upgrading the drive on a ship then sending it somewhere did not result in a faster travel time than it would have without the upgrade.

Yeah I think the travel time between areas only changes with that research option, I've upgraded my starting ship's drive and that still seems to take me the same number of days to travel places as well I think. But I'm curious whether it helps them being able to get from the starting point to a station, or from one station to another in the same area. Or if the game just doesn't simulate that at all and it's all the same how fast the ship is.

Some of the upgrades do improve surviveability. I'm wonder if that matters when not in system.

Upgrades do matter in combat, whether locally or out of sector. They don't affect non-combat situations outside your current sector though.