As you said, the businesses that don't want customers are almost non-existent, so there should be no problem with requiring that businesses serve the public. After all, it's in their best interest to so.
No, businesses do not serve "the public," we serve OUR customers. Our customers are people who pay us, don't abuse our staff, dress appropriately when they enter our business, fit our business model, are profitable, etc. If you enter my business, my staff will decide if you fit our customer base.
As for your standard there should be "no problem with requiring" businesses to serve the public (who we don't serve), let's pass a law by your reasoning requiring flushing the toilet, your standard is "so there should be no problem with requiring" us to do so. Most people do flush the toilet, what's the harm? Actually, there are three problems with it:
1) It's not a legitimate power of government. No one has the right to require legally toilets be flushed just like no one has the right to force someone to build them a cake. Positive rights are an oxymoron, you have no right to compel anyone to do things for you, you only have the right to compel them to not do things to you that you don't want them to do.
2) Frivolous lawsuits. A jackass gets into a party at your house, you kick him out. The next day you get a lawsuit, you didn't flush your toilet. That's why he left your party.
3) General enforcement. Well now, the government wants to make sure you flush your toilet, it's the law, they are just enforcing the law. So now there are more laws to verify you flush your toilet and flushing the toilet is access to your home which then is access to all sorts of other things they want to verify and monitor
And a law requiring flushing the toilet is absolutely as ridiculous as a law requiring anyone to bake a cake. Here's a far easier solution in the incredibly unlikely event that happens. Walk across the street to their competitor...