The timeline fluidity is a bit of a frustration for me as well, despite the fact that I've been reading comic books for 30 years, and it doesn't bother me a whit when I see a flashback to Spider-Man's origin and people have smart phones or whatever. But even trying to reconcile Spock claim in "Space Seed" that the last of Earth's World Wars took place in the 90s, with the implication in "Encounter at Farpoint" that a war was fought with atomic weapons leading up to the mid-21st century.

Part of the problem, obviously, is that they every decided to give anything actual dates. I understand the inclination though; no doubt in 1967 when they were sitting down first build out this world, they wanted to present the idea of WWIII to be looming on the horizon. A lot of the people viewing this when it aired episode would remember being shown videos telling school children to duck under their desks for protection in case of a nuclear attack. But then Voyager travels to 1996 in "Future's End" and there's no indication that humanity just went through the third World War.

Hell, in TOS, continuity wasn't maintained episode to episode for some things. In "Balance of Terror" they hadn't added photon torpedoes to canon yet, so the Enterprise was firing phasers bolts with proximity explosions at the Romulan Bird-of-Prey. There are a handful of different names for the organization the crew serves before Starfleet was named in "Court Martial" and even after that, they still called it Spacefleet Command one last time in "The Squire of Gothos".

Anyways! Obviously I think the subject of continuity in Star Trek is pretty interesting in and of itself, and I would argue that visual continuity aside, the new iterations from Disco onwards have been fairly mindful of canon. There are occasionally things that skirt the line, but very little that actually break continuity. A big one for me is the site-to-site beaming. In "Day of the Dove", Kirk asks if it's ever been done and Spock discusses how dangerous beaming from one point on the ship to another is, and how it requires pinpoint accuracy, but in Disco they do it all the time, and usually just with a voice command to the computer. I also don't like how fast the transporters on Disco and SNW work, but I don't think that rises to the level of a continuity violation.

The idea that the new wave of Trek is lousy with continuity errors is, in my opinion, largely perpetuated by people looking for reasons to dislike the shows, and who don't know canon as well as they believe.