Yea, I am going to have to side with the business. Making a cake is not the same as having to handle someone's genitals. Just as we have specialists in medicine that deal with specific fields, the same could apply here as well. OBGYN's do not take male patients as that is not their specialty. When you wax a female, you do not actually have to handle their genitals as there are no balls or penis to have to move or "lift" out of the way to be able to do the wax on a woman to begin with so you can safely wax a woman without having to touch their genitals at all. On a male body, unless you want hot wax on the penis or balls you would be forced to move them out of the way while applying the wax.
On the religious issue, I do believe one's beliefs should be respected and considered to an extent, but they are not necessarily the only factor to be considered. Many people choose to have a career in jobs that they can also respect their religious beliefs. If they view interacting with someone of the opposite sex's genitalia as being sexual, they have the option to choose jobs where they would not be forced to do that. Giving women waxes is not a job that they would be dealing with male genitals anymore than being an OBGYN would be and nor should they be expected to. There are salons that provide waxing services for male genitalia and they would have to see one of those that specializes in this rather than expect one that does not to perform a service which is outside their field, or find a business or individual who is willing to do so rather than try to force someone who is not to do something against their will.
There is no religion that says you can't bake a cake for someone because you disagree with their beliefs, thus I do not see refusing to make a wedding cake for a gay couple as actually in violation of one's religion. The baker on the other hand can refuse to add anything that they may find offensive to a cake such as refuse to add genitalia, swear words ect. They do however have 100% creative control over what shape they are willing to make a cake and what they will or will not write or decorate a cake with. If they are willing to make a cake of the same shape and/or decorations/ writing for a heterosexual couple, they should also be willing to do so for a gay couple, as that is not actually doing anything that would impact their actual religion and it is not asking them to do anything different than they would do for anyone else. Trying to have someone handle someone else's genitals however is not asking them to be treated as they treat anyone else, it is actually asking them to be treated differently than they treat others, so that is why it is seen as an unreasonable request and if the person who is being asked to do something different for them, they have every right to choose not to.