What gives an Asian the right to be served at a restaurant if it is not the right to be free from ignorant bigotry?
What gives a black child the right to go to the same schools as the white children if not the right to be free from ignorant bigotry and it's harmful effects?
Are you saying black people do not have the right to be free from ignorant bigotry in the market place?
NOBODY has the right to be free from bigotry, or free from hatred, or free from offense.
What you're referencing are rules placed on specific types of services that constrain the behavior of those services. Those services are prohibited from refusing service on the basis of specified characteristics, which include race, sex, sexual orientation, and belief, and in some locations gender identity.
Sometimes there is a conflict between those characteristics. It's not uncommon to have a conflict between belief and sexual orientation, or between belief and sex. Then it's up to the courts to decide which characteristic takes precedence, and in which circumstances the state is permitted to violate one set of rights in favor of another.
In the case of the prior controversy over the baker refusing to bake a cake for a gay wedding, the conflict is between the baker's religious beliefs which are protected under law, and the sexual orientation of the customer which is also protected under law. In which case, the prosecution needs to be able to prove that the baker was discriminating against the characteristic as a whole, rather than against a violation of his specific religious beliefs. Phillips' prior argument was that he was not discriminating against homosexuals across the board, he was discriminating against a specific ceremony or observance, where that specific ceremony or observance is in opposition to his own religious beliefs. He argued that he would have no objection creating a different type of cake for a gay person - a birthday cake, a retirement cake, any number of other types of cakes. He only refused to create a cake celebrating a particular ceremony, and that he would not create a cake celebrating that specific observance for anyone. It gets a bit touchy, because the only people who would want a gay wedding cake are pretty much gay people. But on the other hand, if the gay guy's completely heterosexual mother came in to order the cake, Phillips would have refused to make the cake for her as well - even though she is heterosexual, he was refusing to support the observance for which the cake was being made.
In a similar scenario, in Canada a few years ago there was a conflict between gender identity and either sex or religion, depending on how you look at it. Yaniv identifies as a transwoman, but is physically unaltered. Yaniv wanted to have a Brazilian wax of their genital region. They targeted small businesses, run privately by minority females, predominantly of middle eastern descent. When they refused to provide waxing services, Yaniv sued them. Yaniv put several of them out of business. At the heart of that issue is the conflict between 1) gender identity and sex and 2) gender identity and religion. On the first topic, the business owners were all female, and provided waxing services to 'women' by which they intended 'female'. Yaniv identifies as a woman, but is physically male. The question becomes: does Yaniv's gender identity as a woman entitle them to coerce females to handle their male genitals? That's a question of sexual boundaries, and protections based on sex rather than gender identity. On the second topic, most of the business owners were muslim, and their religion specifically forbids physical contact with males who are not related to them, and sexual contact with any male who is not their spouse. That question then leads to whether or not Yaniv's legal protections on the basis of gender identity override the business owners' legal protections on the basis of their religion. Neither is an easy question to answer.
This case is substantially similar. Phillips' religious views are that sex cannot be changed, and that a gender transition cake is celebrating an event that is in opposition to his religious belief. In the same way that he previously argued, he says that he would not deny Scardina a different type of cake observing an event to which he does not have a religious objection. Because he argues that he would provide a different type of cake to a transgender person with no objections, he argues that he is not discriminating against transgender people as a whole - he is only refusing to provide services in support of a belief-based observance that is in opposition to his own deeply-held beliefs. That then makes this a conflict between the legally protected characteristic of gender identity on the part of Scardina, and the legally protected characteristic of religion on the part of Phillips.
At the end of the day, you are not required to believe that Phillips is in earnest with respect to his claims. You are free to doubt that he would provide other types of cakes to a transgender person. The question really is whether or not the claim of a conflict of protected characteristics is valid in the first place (something for the courts to determine), and if so, which characteristic takes precedence (also for the courts to decide).
My personal opinion is that yes, it is a valid conflict of rights relating to protected characteristics. With respect to the second element, because cakes celebrating gender transition are easily available from other bakers without undue hardship, that it is appropriate for the state to protect Phillips' freedom of religion above Scardina's gender identity. If the majority of bakers held the same belief as Phillips, and it would be difficult for Scardina to obtain a cake celebrating gender transition from any baker, I would support gender identity being protected above religion.