I think having problems with continuity issues in a prequel is a valid concern. If this was Star Trek in the 25th century we would not be having this discussion
No more than any other show. And yes, a 25th Century show would generate continuity concerns.
Just because the other shows had minor mistakes it doesn't mean it's okay for a prequel series set before those shows to impose major continuity errors. It's like saying "he stole an apple so it's okay for me to steal an apple". Being a prequel it had a duty to stick to the facts. No Enterprise NX-01 played any part in the on screen history of Trek until 2001 and no amount of retconning and backwards bending is going to change that
The "facts" are what ever pops up on screen. That includes new information from new installments. Spock doesn't have a brother, until he does. Kirk didn't date Carol Marcus, until he did (In fact Carol Marcus is never mention in TOS. I guess she's not as important to Jim as Ruth

). There isn't a ringship Enterprise, until there is. There isn't an NX-01, until there is. That's how fiction works. Stuff is constantly being added to the "past".
Nothing after the explosion of Romulus and Spock's disappearance is written in stone as far as the Prime Universe goes. In 2001 Trek began with TOS and ended with Voyager's finale. The future of the franchise had not been written yet and still hasn't been apart from the Abrams movies set in a different universe.
And nothing prior to the Cage was written in stone until Enterprise.
The thread asks the question "what is the worst continuity error in Star Trek history?".
My answer is Enterprise, a prequel full of continuity errors.
Nah. The only one that ever held up to scrutiny was the cloaking device in "Minefield". A better record than the other shows.
Riker and Troi casually using the holodeck during "The Pegasus" as if Riker's entire career isn't about to be ripped away from him. Their demeanours in this episode are completely wrong as is the timeframe available to them to take part in this silly escapade.
Not sure what the continuity error is here. TATV is a horrible episode. But that's not a continuity issue.
Phlox finding a cure for Borg assimilation. The Borg showing up on the NX-01. Not one person thought to "google" that incident in the 24th century? Janeway can look up an unimportant captain's log from 100 years earlier but Picard can't look up this very Borg-like incident from before Q Who?
Since Earth and the NX-01 weren't attacked by "The Borg" why would Picard be googling the Borg? The facts are the Borg (or at least a culture of cyborgs) have been a known threat for decades prior to "Q Who". ("Scorpion")The Hansons must have gotten their information from the NX-01 logs and the accounts of the 23rd Century El-Alurian refugees. (Generations) Then there are Romulan records. ("The Neutral Zone") So,"The Borg" are part of the historical record going back Centuries.
Enterprise builds on the current state of continuity of the Borg. Looking at the episode, Picard went to Guinan, who had first hand experience with the Borg, rather than a computer,which doesn't mean that information isn't available. It also makes more sense dramatically.
As for Phlox's cure, three words: The Borg adapt.
NEXT!
Trip failed to mention the aliens he encountered had enormous ears and were obsessed with profit? Hmmm. Who does that sound like?
Who says he didn't? There was no visual record of a race call "the Ferengi" available to the E-D. (kind of the point of the episode) The profit driven, big eared aliens encountered by the NX-01 were not called "the Ferengi". No reason to link the two, until DaiMon Tarr appears on screen. From that point on the profit driven, big eared aliens are the Ferengi. That's how xeno-anthropology works.
NEXT!
The Defiant, complete with state of the art 23rd century machinery, arrives in the 22nd century of the Mirror Univese. Over 100 years later technology has not progressed at all by the time of Mirror Mirror?
Cultural and technological stagnation are not unknown in the real world.
More anthropology.
What else you got?