One could set up a dress code or any other arbitrary policy as a proxy to refuse entry to any group, without explicitly doing so.
I suspect you're not happy with that either, and that's my point.
Not in the US you can't. If a policy, even when applied equally, unduly affects a protected class, it's unconsitutional.
For example, if a restaurant enacts a "no headwear" policy it's still unconstitutional because by and large this is primarily going to affect the muslim population (and a segment of the jewish population), but will have very little effect on anyone else.
If the policy was instead "no headwear with words on it" the policy would NOT be unconstitutional because it does NOT unduly affect a specific group of people.
There are other exceptions, of course, generally around security. You can imagine a bank not allowing the full covering of ones face for security reasons even if that does appear to target people who wear burka's day to day.
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so long story short is that it's not super simple, but in the US you absolutely cannot try and get around it by proxy and there is established law on how to identify this.
I don't care about the US, and I don't think the laws there should be an example for the rest of the world.