I’m sure there’s a deep reason why a ba’al mum specifically can’t do the job, but in terms of “fairness”, which really seems to be her question, the assumption is wrong. Everything we have, including oppurtunities to serve Hashem, is really a gift from Him, and it’s not reflective of proper hakaras hatov to ask why we don’t receive even more.
There’s no word in Loshon Hakodesh for “fair”, because the concept is not real.
We perceive our existance as a given, so we think that yashrus, justice, demands equality. In reality, our very lives are matnos chinam, free, undeserved gifts, so if His wisdom sees fit to create people with different jobs, we are silly to complain. A cow has no right to complain that he’s not a person.
I’m sure there’s some inherent flaw in the service performed by a ba’al mum. The same is obviously true for a flawed animal, which is passuk (disqualified) for a korbon (sacrifice). The disqualified animal has no more right to complain that he can’t serve Hashem as a korbon than he has a right to complain that he’s not a person.
Similarly, a Kohen who has a mum is not being discriminated based on his blameless mum, but for a reason only Hashem knows, the job to perform avodah was not granted to him.d