Of course they can, restaurants enforce dress codes all the time. No tie? Get out.
Cops? You’d have to be a special kind of stupid to think a cop on duty with official business can’t go anywhere with his weapon.
And as far as your last point, Yes, you can set standards if you’re the owner but you can’t discriminate by the usual things or violate the ADA doing it. Of course you’re shooting yourself in the foot by doing so, demanding all men where pants to enter the store is actually not illegal, but it’s a guaranteed to close your doors.
Go into a upscale diamond cutter/ designer outlet wearing flip flops and cutoffs, you’ll be asked to leave before you take a step in there, and they are a public outlet. I know this because I saw it happen 33 years ago buying my wife’s designer wedding ring.
That is because here is no constitutional right to wear or not wear a particular kind of clothes in a restaurant. However, if the restaurant has a dress code which does cause actual constitutional rights to be impacted then the restaurant cannot enforce that aspect of the dress code.
A dress code which says ties required for men will probably pass scrutiny. A policy which says "no yarmulkes" is clearly illegal.
> I know this because I saw it happen 33 years ago buying my wife’s designer wedding ring.
33 years ago is not today. I went into an upscale jewelry store in Los Altos California in garb similar to your description ( a scruffy T-Shirt printed with “keep calm and nuke it from orbit”) and enjoyed a polite experience.
In Silicon Valley billionaires routinely wander about about by themselves wearing scruffs, bad hair, and drive dirty cars; as long as you’re polite and don’t smell too bad they’re usually willing to hedge on premise you might be able to pull $100k out of your backpack and buy their ridiculously priced merch with minimal haggle (time is more valuable than money). The really nice places don’t have a storefront and only do business with people they know or through introduction, but odds are high you could show up in 2 coconut half shells and a grass skirt and they wouldn’t blink an eye.