No the X-men do not need their own universe and the reason they aren't received like the Avengers is because X-men is based on the dynamics of racism, which is where you get the Magneto (Malcom X), Xavier (MLK) parallels.
The Avengers are a mainstream, well known outfit who are mostly human's who have been enhanced or transformed by science and technology. The main ones are public figures and even government operatives at this point in the story and the only one who really has a secret identity is Peter.
The two Avengers who are the least human also happen to be the most hated/feared...especially Wanda who is doing shyt beyond the realm of some fancy gadgets or potions. They don't even know she's a mutant yet and are already shook of her, how you think they gonna react when they find out there are people like her BORN that way...HIDING who they are, what they're capable of and what their moral compass is like?
Being that Wanda's powers are already tied to the infinity stones, anybody on the planet who had X-Genes lying dormant could have been activated by the snaps. The mutants in the comics stay on the low for a reason...the ones that could already exist will be finding it increasingly harder to stay hidden as news stories come out about people being 'blipped' back to reality with newly acquired talents that they can't control.
I'm assuming the WandaVision TV show on Disney + will do some sort of variation of House of M to click everything into place, she's going to be the key to integrating the X-men into the MCU.