Should mask mandates apply to the vaccinated?

If vaccines are so effective, then why are vaccinated individuals required to wear masks? Most of my criticisms of our handling the pandemic are more likely to tick off Democrats. In this case, I'm going to go in the opposite direction and criticize something that I heard primarily from Republicans. If you are going to mandate masks, you should not make an exception for people who are vaccinated. Although I believe the effectiveness of vaccines was overstated, I do not believe that a lack of an exemption proves a lack of confidence in the vaccines.

Before I go any further, let me quickly recap my views on mandates. I strictly oppose the government imposing mandates on businesses and organizations. I also do not believe that people should be required to carry paperwork with them just to live their lives. If a business wants to mandate masks in their facilities, I'm perfectly fine with that.

In the case of mask mandates and exemptions for vaccinated individuals, how do enforce that? How can you be sure that everyone you see without a mask has been vaccinated? You could always ask for proof of vaccination each time, but this would result in these requests becoming excessive. Either employees would actively harass customers who are doing no wrong, or these employees would reduce their effort to the point of being ineffective.

These businesses could always go by the honor system, but is that really a mandate? People would just pretend to be vaccinated so they wouldn't have to wear a mask. There are also people who might assume mask mandates have been lifted because they are seeing others without masks. Additionally, some unvaccinated individuals would likely become used to this and enter more strict businesses without a mask. This can increase the burden on such business while people can be exposed to those who are unmasked and unvaccinated before they can be asked to mask up.

I do not want to actively require individuals to prove their vaccination status. Even if vaccines were 100% effective in stopping the spread of a virus, I wouldn't want to provide an exemption to mask mandates. I don't see any good way to enforce these mandates alongside such exemptions.