I'll grant that a dojo can make accomodations for an individual - we do that all the time. We have people who can't perform suwari waza because of bad knees and so forth. The bowing to the shomen at the start and end of class is another simple example. I've seen people who didn't and that wasn't a problem. Where this becomes a big deal to me is when an individual won't train with other dojo members on the basis of gender. That's where the world becomes a damned slippery slope in my estimation - if I'm a fundamentalist Christian, would it be acceptable to refuse to train with a gay or lesbian member? An unreconstructed southerner who refuses to train with african-americans? A conservative Republican who refuses to train with a leftie anarchist? My answer is simply no, it isn't acceptable and is discrimination in any of those cases. I would choose not to be a member of a dojo that allowed or encouraged that kind of conduct. In the same manner, those who are adamant about their beliefs to the contrary can choose to train somewhere where they can maintain the integrity of their beliefs. I think we can treat each other with respect and can live very well on either side of the threshold to a dojo not of our liking.
Some of your examples make no sense. Political and racial preferences are not the same thing as religious preferences. I amy not agree with your religion, but I would respect it. So if you were a practicing Christian and didn't want to practice with someone you knew was gay, then that would be your choice. Why? Just like gay people want to be treated like everyone else and respected as people and want you to respect their choice of lifestyle, then likewise, it is not unreasonable for them to do the same. Funny thing is, if men decided to open a dojo for men only, as to avoid such a problem as this, then someone else would have a problem with that.