Sometimes, the reason that this isn't an option is that you don't have that much control over what data is provided. Suppose, for example, you want a fancy AI that reads a Résumé and filters on suitability for a job. There isn't a particularly rigid formula about what people put in their Résumé, which makes it difficult to exclude things you'd rather not consider.
Where you do have more control about exactly what information you consider, it can still be thwarted by correlations. Think, for a moment, how this pans out with a human decision maker. You want to ensure that Joe Sexist gives women a fair chance at being hired, so you make sure that there isn't a gender field in the application form. You also blind out the applicant's name, since there is no good reason that a name should determine suitability for a role, and including it would reveal a lot of genders. But you don't block out the hobbies, clubs and societies entry, because it's thought to say something positive about an applicant if they were the captain of their college sports team. Joe Sexist, however, considers it a positive if an applicant captained a male dominated team such as American football, but considers negative being captain of a female dominated team! Some might say that wouldn't quite be bias against women; it's bias against players of sports that Joe Sexist considers effeminate. But really a skunk by any other name would stink as bad.
The same sort of thing can happen with AI. Now to be clear, the AI is not sexist. It is a blank sheet with no preconceptions until it gets fed data. But when it gets fed data, it will find patterns in the same way. The dataset it gets given is years of hiring decisions by Joe Sexist. As suggested, there is no entry for gender, but there are fields for all the things that might be considered slightly relevent. For example, we include whether they have a clean driving license. The AI notices that there is a positive correlation between the number of road traffic offences an applicant has and Joe's likelihood of hiring them (Because, of course, there happens to be a gender correlation between dangerous driving and gender). Again, the AI has no preconceptions. It doesn't know that traffic offences are dangerous and should be weighted against. As far as its dataset suggests, they're points! With this sort of information in a dataset, the AI can exhibit all the same sorts of biases as Joe Sexist, even though it doesn't know what a "woman" is!
To expand this with specific numbers, suppose that your dataset has 1000 male and 1000 female applicants for a total of 1000 places. Of those, 400 of the men and 100 of the women have a tarnished traffic record.
Joe Sexist was not in favour of reckless drivers: in fact a clean traffic record guaranteed you would beat an equivalent candidate with a tarnished record. But he was very in favour of men: being male made you 9 times more likely to get hired than being female.
So he gives places to 900 of the men: all 600 of the clean drivers and 300 dirty drivers.
He gives places to 100 of the women: all to clean drivers.
Now, you take away any mention of gender in the dataset.
There are 2000 people, 500 drive badly, 1500 drive well.
Of these, 300 bad drivers get jobs, and 700 good drivers get jobs.
Therefore the 25% of the population who drive badly get 30% of the jobs, which means (as far as an AI that just looks blindly at the numbers is concerned) that driving badly suggests you should get the job. That's a problem.
Further, suppose you have a new batch of 2000 applicants with the same ratios and it's the AI's turn to decide. Now often AIs actually make this even worse by exagerating the significance of subtle indicators, but let's suppose that this one does everything in strict proportionality. The AI has learned that 60% (300 / 500) of the bad drivers should get the job. It doesn't know about gender, so it at least allocates the bad driver bonus "fairly": 240 male and 60 female bad drivers get jobs. Then 280 male and 420 female good drivers get jobs. This comes to 520 male and 480 female applicants getting in. Even though the original applicant pool was balanced and if anything women were better (at least at driving) the original sexism in the training dataset still gives some advantage to the men. (as well as giving an advantage to bad drivers)
Now, don't let me completely disuade you. In the human case, it is a known fact that blinding out some information does indeed give more balanced hiring decisions. And even in my toy example, while it doesn't get to fairness it has massively reduced the scale of the sexism. So yes, it probably would make the AI somewhat less sexist if the most blatant indicators aren't provided in the dataset. But perhaps this gives some intuition about why it's not a complete solution to the problem. There is some sexism that leaks through, and it also causes the system to make very weird associations with other bits of the dataset.