Couldn't the elevator be the loading screen. Like a snapshot of your character standing in the elevator with the lights moving in the panels. To make it look like the elevators moving like the show.
If the loading screens were the ship you're flying moving at warp, like the shows. Warping in or out. The loading screens for ground were you and your crew beaming out then beaming in they wouldn't be so bad.
Those generic loading screens with things that almost look like LCARS, are immersion breakers. They remind you're playing and game and you are waiting on it to load.
There are way to many loading screens. Loading screen to start the mission, then the space part. Loading screen to beam down and do the ground part. A loading screen to go up and finish the last space segment. Then another loading screen to leave.
This is the part of the game where the engine is the problem. Because you can't be both you are either your ship or your character. So you have to wait to change. You can't be your character on your ship it just a instance.
Have an elevator animation that plays on top of the loading screen. You can clearly do animations given the current loading screens, even if they're a bit choppy. So having a shot of some starfleet dudes in a turbolift with elevator music playing while the loading is going on would be pretty snazzy.
I've seen several other games that actually do stuff instead of having a loading screen (but might as well be loading screens), so it should be possible to do here. I guess it's a resources issue.
I have some Foundry test stuff with one. It's sneaky though. What you do is layer 100 or so identical objects at 1/10 of a meter height difference and use reach triggers to remove and add them to create the impression of an elevator moving.