Since Star Trek: The Next Generation took off in 1987, there hasn't been a Star Trek series that hasn't had some type of connection to The Original Series, even if it was by a thin thread. Star Trek: The Next Generation was for the new crew aboard the Enterprise, which was the orginal starship. Star Trek: Deep Space Nine was connected to The Next Generation through Chief O'Brien and the Battle of Wolf 359, which killed Commander Sisko's wife, Jennifer. Captain Jean-Luc Picard had been assimilated by the Borg at the time and was involved in the death of over eleven thousand people. The bond was strengthened when Lt. Commander Worf came to Deep Space Nine.
Star Trek: Voyager crew members visited Quark's Bar on Deep Space Nine before their voyage began, and Reginald Barclay, who had served aboard the Enterprise on The Next Generation, showed up in several episodes as did Counselor Deanna Troi. Then, of course, there was Star Trek: Enterprise which took us back to the first ship that started the franchise. And these are just some of the smaller ways they were connected.
Since Star Trek: Enterprise ended in 2005, every series that followed has had a connection of some type to Captain Kirk, his crew, and/or the Enterprise. Screenrant has suggested a new Paramount+ movie with an entirely new crew "without any ties to pre-existing characters or canon," that would "create an entry point for new fans."
It's certainly something that should be considered, and with all of the alternate timelines and centuries available that haven't been explored, a new Star Trek series or movie could find viewers who are looking for an interesting science fiction show even if they haven't seen one episode of Star Trek.
One might argue that there will always be a connection to Star Trek: The Original Series because it was the prime show. It introduced Starfleet, the Federation, and the Prime Directive. But that tie doesn't have to include any ties to the Starfleet officers that have gone before them. So basically, it would still be Star Trek but set in a different part of the galaxy. Star Trek: Discovery tried to do this, but the producers fell back on the tried and true by connecting Michael Burnham to Spock as his adopted sister.
So what about a series or a movie that has no legacy characters, no references to previous battles or starships, and is about a crew of Starfleet unknowns heading into the galaxy for the first time? This could be a chance for a new captain to break ground, perhaps as a new alien species.
Though there will always be some strings that unite these shows, there are plenty of ways to allow a new series or movie to venture out on its own without having to exist in the shadow of the series that have gone before.