I'm somewhat surprised by the number of people trying to see this as a legal problem only. Don't forget that an advisor/advisee relationship is very special. Taking a PhD student under your wings is not entirely unlike adopting a child (if only temporarily), and rejecting an interested PhD student is sometimes the responsible thing to do, when there are warning signs flashing.
I don't think it is a wide spread practice that potential advisors have the obligation to take anyone meeting certain qualifications. In fact, I rather think the opposite is true.
- A potential advisor can reject such a request for very personal reasons. May be they have interacted with this grad student earlier and didn't like what they saw? May be they have work for just one grad student, and already have their eyes on someone else who they think could do a better job, and now came up with a lame excuse. May be Dr X recently lost a loved one to kidney problems, and cannot bear meeting with a living reminder daily?
- Another plausible scenario is that may be Dr X heard about Joe being late to take his quals, and therefore was predisposed negatively? When hearing about the true reason for the delay, they were simply unable to do a mental U-turn, and blurted out something that made them look like an ass. Not nice, definitely. But the kind of s**t that can easily happen, when you are put on the spot and need to justify the decision to decline a particular grad student.
We simply did not hear the whole story.
This is math, meaning that the PhD students don't often get to work with teams. If there is an advisor/advisee chemistry problem, a rejection will save both Dr X and Joe a lof of pain further down the road.
- Joe can (should?) talk with the Director of Graduate Studies at the department (if such a resource exists). They know the local circumstances and can give advice. I don't think a DGS would do much, but if similar stories about Dr X pile up, then there may be something actionable, and the DGS will have the means to do something about it. The DGS is not likely to gossip and tell that Dr X has a reputation of being a difficult person to work with (other grad students are there for such things), but it is their job to listen.
- If Dr X discards talent left and right, they are shooting themselves in the foot, and may have difficulties recruiting in the future.
- Similarly, if Joe goes public, he has to live with the consequences. Bad advice IMHO.
Joe should just continue shopping for an advisor unless he already has done so.