Candidates partially compete through their networks. Hillary Clinton spent years developing the support that carried her past Bernie Sanders. Donald Trump's path was different, but he too relied on his established brand. It seems unlikely that the existing candidates, who are accustomed to competing in this way, will give way to this new system that works differently. And by definition, these are the candidates with established networks.
Another issue is whether it is practical to hide identity this way. Look at the show The Masked Singer. In that show, contestants hide their identity, as you suggest. But people speculate, often correctly, on who they are. For example, I know that the Alien was correctly identified before her identity was revealed. As was the Raven before her. How do I know? I read those predictions before the shows and saw them proved right in the after-show reports. Why would it be more successful in politics than in a reality show?
What happens if someone simply reveals that they are running? What happens if a supporter does? An opponent? I can easily see a court case that establishes that it is a candidate's right to reveal identity, overturning any rule against it.
How would this work with the existing ballot system? Currently candidates ask to be on the ballot (in the United States). But under this system, you couldn't ask that. You couldn't go to voters and tell them who they are and ask for signatures.
What happens to experience? Successful presidents were often successful governors or senators or generals. But under this system, we wouldn't know the candidate's experience. And we can't check the candidate's past positions or truthfulness or, well, anything. You aren't just hiding gender and race. You're hiding their past identity.
If you don't hide that identity, then you're not hiding their race or gender either. E.g. in 1980, a retired actor who was formerly the head of the screen actors guild is running for president after almost winning the nomination in 1976 and being governor of California for eight years. How would people not recognize Ronald Reagan? And of course, he was known to be a white male.
In 2016, a billionaire businessperson who has also appeared on reality television is running. Against a former first spouse.
In 2012, a multi-millionaire financier who had been governor of Massachusetts is running.
In 2008, a former community organizer from Chicago who is now Senator from Illinois. Is running against a former Marine famous for being tortured while held as a prisoner of war, who is also a current Senator.
In 2004, a Senator from Massachusetts who had previously been a decorated military officer is running.
Does anyone seriously think that these people would win their first office without these aspects of their biography? Not to mention that politicians have been known to exaggerate their biographies. How would you check?
If it worked, it might remove bias. But it would also remove discernment. Unless everyone did this, all the time, candidates couldn't give enough of their background to be considered. And I don't think it is really feasible for everyone to do this to eliminate bias. How would people date?
An easier way to achieve the same objective would be sortition. Just draft the right proportions of each group. It's a mechanical process, so it can only be biased if the process is. And it would save the problem of anonymity. You still lose discernment, but you gain representativeness.