The thing is, about 74.5% of Trek plots would simply be gone if our heroes couldn't see what was happening around them in real time. No deviating from course to examine an odd reading, a shipwreck, a planet that didn't exist two days ago. No way to track a fugitive. No chance to stop an invader. And no way to contact home base for interesting new assignments (unless one postulated a system that can be used for realtime communications but not for realtime sensing, which would exhaust the salt deposits of Utah).
No, the same plots would exist, they would just have to be established with a bit of forethought instead of a random "sensors are picking up a planet that wasn't there two days ago." A really great example of this is "The Corbomite Maneuver" where the destruction of the Marker Buoy has a brief interlude several minutes long during which Kirk is able to go below, grab a sandwich and chat with McCoy before the Fesarius actually catches up with them. Had this been, say, a Voyager episode, Fesarius would have right on top of them after 15 seconds of dramatic music.
In such cases, starships would either have to rely more on long range probes (whose subspace signals can travel faster than light) or writers would have to take care to build the passage of time into the script itself instead of compressing the entire event into a single scene (much like the "Hail them." "No Response" thing). The time it takes for your science officer to receive back the scanning pulse is time that could be spent on strategizing, looking up records, or even -- GASP -- character development!
OTOH, those stories may not require instant cognition anyway, they would merely require reducing the size of the Trek universe so that "We're the only ship in the area" really just means "there's another ship in a neighboring system, but they're too far away to help."
Nearly everything about Trek's futuro-tech is "dependent on plot needs" anyway, including warp drive, Starfleet regulations, Federation political and economical structure, interstellar alliances and animosities, and the biology of the heroes and villains. There's no inherent downside to that. All it takes is a bit of bookkeeping.
The inherent downside is that the story elements themselves become unimportant window dressing to what is essentially a sci-fi morality play (e.g. Gangster Planet, Black and White guys, Yangs vs. Kohms, etc). If the background and scenery matters at all, it needs to be set it stone first and not shifted around all the time for writers' convenience. If it DOESN'T matter, then there's no real point to having each episode take place on the same ship with the same characters in the same fictional universe; Star Trek becomes a fancier space-based "The Outer Limits."
If you're going to have a consistent setting over a number of years, the most commonly recurring plot devices need to be set in stone. Not just for the audience, but for the writers; in later years in TNG they ran into situations where even the writers couldn't remember how half their technology worked and wound up either contradicting earlier stories or pulling technobabble out of thin air to serve as temporary plot devices.