Obviously a Christian is not going to refuse to serve sinners. They'd have no customers. But they can refuse to take part in sin.
No one asked them to participate in a wedding... just make a cake. If the law attempted to compel Christian ministers to solemnize gay marriages, that would clearly be a different thing. Whether a Christian wedding photographer could turn down a job to photograph a gay wedding is a tougher call, since the photographer would actually have to be at the ceremony. (Aside: I personally swallowed my objections and photographed my brother in law's wedding to his partner, but I wouldn't criticize someone who made a different choice. As for my brother-in-law and his partner, I disagree with their lifestyle, but like both of them quite a bit.)
why can't white supremacist businesses refuse to serve blacks, or male businesses refuse to serve women?
I don't know. We have essentially eliminated the right to freedom of association. It might not be so bad if we were a more homogenous culture, but now that the policy is "invite the world" I don't know how that works in the long run when you are basically forced to interact with every culture in the world.
You just have to be tolerant. That doesn't require you to participate beyond doing business. And if your business requires more participation than you are comfortable with, then that business isn't for you. I couldn't be a sports photographer, since too many games are on Sundays, for example.
This may well mean that some Christians find that the wedding cake business is not right for them, just as observant Jews don't take jobs or run businesses that require them to work on Saturday, devout Mormons don't work in bars, etc.
Doesn't this kill culture, though? "Sure, you can keep your religion, but you can't actually live according to its teachings, and you can't segregate into a place where you can?"
Meh. That's just reality, and it doesn't do anything to "kill culture". I'm a Mormon, and I work for a company (or at least on a team within a company) where drinking is a huge part of socialization. I'm a bit uncomfortable with the booze-heavy social outings, and it's not unlikely that some of my teammates are a little uncomfortable with the fact that I'm not drinking. They respect my choices, though, and I respect theirs. It works.
An excellent example of accommodation is the long-standing set of adjustments that people who interact with the Amish make, as well as the Amish themselves. There's a dramatically different culture in many ways, and one that has many friction points with the larger community, but it works. Not perfectly, but well enough for everyone.
This all seems like an extremely low entropy arrangement. We have to pump a massive amount of energy into coercing every combination of race, religion, culture, gender into equivalent cogs in society.
This is a gross mischaracterization. There's no "massive amount of energy" involved, and no implication that everyone must be equivalent cogs. All that's required is a little accommodation and a willingness to allow people to be different.