As @Buffy comments, the other answers do not seem so accurate about R1 and R2 grad math departments in the U.S. That is, funding is mostly as Teaching Assistants, administered through the department, not by individuals, and funding-and-admission decisions are also made at a department level, by a committee.
Yes, having a faculty person ardently promoting your case to the admissions committee is definitely a good thing, but is not necessarily decisive, depending on that faculty person's record of following through (or not), mentoring, etc.
Yes, asking faculty ... not whether they "have a spot", but... whether they are currently taking PhD students can be a relevant question. Both people with lots of students already, or who aim to retire very soon, might say that they are not currently taking more PhD/research students. It would obviously be useful to know that, rather than show up and be negatively surprised.
In any case, as other people have commented, do NOT write generic emails with a huge bcc list. Take the trouble to address people by name (!!!), at the very least! And if you describe your own interest as X, which is not much connected to what the faculty person does, don't email them at all.
True, people seem to often "reckon" that sending inaccurate emails to people is harmless, because they can just delete them. It's not that I myself keep a grudge list of people who've sent me spammish emails... but it is fairly antisocial to spam people, in my opinion.