The X-Men don't work in the MCU because there is no way to incorporate their overarching story where the MCU heroes don't look either ineffective in combating or complicit in prejudice.
The closest thing we have is The Hulk, and the only reason he gets a pass is because he palls around with the Avengers. If you bring in the X-Men you basically are just redoing Civil War but on a much larger scale and with clear lines about who the villains are.