I'd argue something else. Maybe just maybe shows like Star Trek worked and there was less picking them apart because they had more interesting characters and entertaining episodes that plotholes were overlooked or ignored. I Believe That people are more critical and pick things apart when they're not being entertained and all they have to do is dissect and analyze every little thing
Although I tend to feel shows were a tighter in their writing than they are now where now they feel like the endpoint is more important than plot logic
Like, say, a ship that can turn invisible nevertheless being tracked for the entire episode, while a ship that can't turn invisible 'hides' as a radar ghost but also sends out random messages that are completely unnecessary and should logically give up the game, but somehow magically don't?
Or how about a man repeatedly described as having no emotion who still smiles and laughs all the time?
Or a professional explorer with extensive safety training who introduces his entire ship to a terrible space virus because he decided for no reason at all to just take off his hazmat glove and start touching things on a ship that had been destroyed by a terrible space virus? Or an episode that's all about overcoming a terrible space virus in time to save the ship from sinking into and burning up in a planet's atmosphere that suddenly turns into a time travel event for no reason?
Or an episode about a superpowerful alien race that's really friendly and only wants to 'test' your 'intentions', even though they very clearly almost destroyed your ship with their first test and quite possibly might have with the second, as well, but no one has any problem with that whatsoever.
Or when a ship finds a strange recreation planet where all their passing thoughts come true and spends most of the episode panicking about the dangers and insanity of it all only for the groundskeeper to pop up at the last minute and claim 'we're sorry, we didn't know you thought this was real. Now that you've panicked yourselves half to death, we'll explain how it works and why you dont need to be afraid.'
Star Trek was no tighter in writing than anything today, even in many of its best episdes. And if you really think people haven't picked it apart just as much as modern shows, you're frankly insane. It's not the amount of picking apart that has changed, it's only the visibility/scale of it because now people from all over the world can come together and share their nitpicks in one place.