I think people take Mystique's speech too literal...
she was trying to inspire and motivate the young group (as part of her story arc of her accepting an embracing her role as a mutant role-model, that she has been running from) so she made the story more relatable to them, by claiming she was the same age as them, and suggesting that the team actually had a name to make them seem more legit then they actually were... it doesn't mean things happened the exact way she described
telling the truth of the matter probably would have traumatized them lol
"I was like in my 30s an scared sh**less, we weren't really a team, we had no name, we all almost died on our first mission which also happened to be our last mission together as I then joined the bad guys while Xavier lied on the beach paralyzed"... isn't really a good motivational speech lol