I don't want them to interact with the existing heroes on their first outing. Marvel should build them up to be the veteran team that they ought to be. Introduce their world the way they did Kamar-taj.
We've been around the Xmen for a good number of movies already so it should not take a long exposition to re-introduce them.
The GoTG did so well in introducing each member. Did we need to follow them from the beginning individually? No.
Firstly see my recent thread for why I think it's possible to use some of the same actors but also give the X-Men a new story in the MCU rather than having to be one or the other.
I think you're absolutely right about the lack of need for exposition regarding individual characters' origins, but we also need an "origin" story that gives the X-Men chance to establish their MCU backstory and also give them a catch-up with regards to the size of their roster and their interaction with the wider super-powered community.
Here's how i think it should go (
which thinking about it is pretty much how I would have liked to see the original X-Men movie go)...
Professor X (I'd go with a new actor, older than McAvoy, younger than Stewart, and more like the latter than the former)
...recruits a new team of...
Storm (Alexandra Shipp)
Nightcrawler (Kodi Smit-McPhee)
Colossus (Stefan Kapicic)
Banshee (Caleb Landry-Jones would be great to see again but they'd have to do the Irish accent whoever it was)
Sunfire (?)
Wolverine (Pablo Schreiber sounds up for it) - previously a member of a Canadian organisation called Alpha Flight, less public than the Avengers but known to SHIELD
(I left out Thunderbird as The Gifted might keep going)
...to rescue the missing "First Class" of...
Cyclops (Tye Sheridan) - with reference to him having a YOUNGER brother called Alex, not the Havok we saw in the last batch of films
Jean Grey (Sophie Turner)
Angel (?) - without the events of Apocalypse having turned him into Archangel, and possibly played by a new actor playing a less rough-and-ready Warren
Ice-Man (?) - A new younger version, being the same age as the MCU Peter Parker, getting ready for an "Amazing Friends" crossover scenario
...and assistant school head...
Beast (Nicholas Hoult)
...from Krakoa the Living Island, pursued by and in other cases encountering by chance, other expeditions to the location including characters who have history with Xavier and/or his new recruits...
...Magneto (Fassbender?) & Mystique (Lawrence?) gradually recruit Juggernought, Black Tom Cassidy, Sabretooth and Silver Samurai (done properly) based on their grudges with Xavier, Banshee, Wolverine and Sunfire respectively, after initially starting out with maybe Destiny & Rogue and a couple more like Toad, Pyro, Avalanche and the version of Quicksilver recognisable from the Fox Universe (
we could later have it explained that Wanda and Pietro were Magneto's children and their Mother was Inhuman, and that Peter is not Pietro, he's their half-brother, that would be a nice retcon and it would need referencing somehow if we got a mutant Quicksilver in the MCU)...
...whilst the X-Men team up with, after battling them first, Wolverine's former team-mates Alpha Flight who want to take him back to Canada after helping to rescue the "original five" X-Men...
... We could get a recognisable take on the X-Men, with retconned backstory allowing familiar characters to pop up in the MCU and others to be given a total reboot, that has it's story roots in the classic storyline that gave us the first major expansion in the X-Men's roster, and throws in their earliest established adversaries from the comic line to quickly and neatly give us a status quo where its accepted that there is now version of the X-Men in the MCU that the audience have an established affinity with but can accept as something new.
By the start of a second movie I'd have the "original five" of Beast, Cyclops, Jean Grey, Angel, & Ice-Man now set up as X-Factor investigations, with Banshee now assistant head of the school, Sunfire back in Japan, and Wolverine AWOL (off doing a solo movie that features Sabretooth, Wendigo, Hulk and Sasquatch and later acquiring Jubilee as a sidekick) leaving an active field team of Storm, Colossus, Nightcrawler joined by Kitty Pryde and either Dazzler&Longshot, Bishop&Psylocke or Rogue&Gambit
...third movie we get Blue & Gold teams