No. Becuase that information would've filled Simba with hatred and desire of revenge towards his uncle, instead of with courage and care for his pride, home and his place in the cerchio of life.
I believe that's why Mufasa only let Simba know that no matter what happened in the past he still loved his son and wanted him to take his place as the rightful king and for him to be happy, and encouraged Simba to face his demons and past no matter how scary it might be. Nothing else mattered at the moment. Eventually, I'm sure he would've lifted SImba's false guilt da letting him know the truth, if Scar hadn't.
I agree with everyone else. I do believe that he needed to figure this out on his own because everyone knows the main character will eventually come to realize what's going on. It's like in video games and stuff.