Admittedly France has tried this bullcrap with burkas before, but that's not exactly something anyone should be emulating, I think we'd pretty much all agree that "I'm sorry but for security reasons you can't buy groceries wearing a burka" is not an acceptable argument. Security doesn't grant free license to override other people's rights.
Bear in mind that the actual real-world examples of the argument "people shouldn't be able to wear masks in stores because of security risks" have for the most part mostly been examples of security being used as a justification to infringe on religious rights or to block marginalized/disabled people from taking reasonable safety measures to protect themselves from infectious disease.
If you're going to bring up an example of security overriding other concerns, at least bring up an example where security hasn't observably immediately become a slippery slope to infringing on people's rights and excluding them from society. Is "stores can ban you for wearing a mask" supposed to make me more comfortable with websites fingerprinting me? I mean, I know where that argument ends up in the real world, it never ends with balaclavas, we've had that argument in the real world and where it actually ends is with immunocompromised people not being able to buy groceries.
So I'm not sure any of this is really supporting your point. Anonymity should not be punished in physical or virtual spaces, and there are huge debates about de-anonymization, facial recognition, and tracking in both public and private physical spaces and for the most part we don't accept security as a justification for de-anonymization.
