They probably thought it was ok since Xavier and Mystique never actually are in the same room in the OT or share screentime.
DOFP was the splitting point that was meant to change her into that OT character.
Mystique and Xavier/Professor X have no relationship in the original trilogy for a reason: timeline has been altered.
According to my scenario:
Sebastian Shaw (Kevin Bacon) was a time traveler. He was born several years after the events of "The Last Stand", in the original timeline (X-Men Origins: Wolverine; X-Men; X2; The Last Stand).
At a certain point, when he was 50 or so, he traveled back to 1912 A.C. and impacted the timeline over the years in various ways... until creating the Hellfire Club.
He also changed Magneto's life in 1944.
The events of "First Class" were deeply caused by him, as well as having many birthdays shifting in time and characters being born later or before (Angel, Jean Grey).
I guess he influenced many powerful people over the decades, impacting their lives - Warren Worthington II (Angel's father) was among them.
FC Moira was the grandmother of LS Moira, as simple as that.
Emma Frost was already existing. Maybe, in the original timeline, Emma Silverfox was a clone of her who was adopted by Kayla's family (that's it).
So in the end, we got the
second timeline:
1962- First Class
1963- Magneto is arrested
1973- Mystique kills Bolivar Trask
1983- Alternate version of "X-Men Origins"
2000- Alternate version of "X-Men" and "X2"
2001- Alternate version of "The Last Stand"
2008- The Wolverine
2023- Post-apocalyptic future of "Days of Future Past"