It seems there’s a decent bit of talking past each other going on, IMO. The conservative criticism of safe spaces generally follows two tracks with varying ranges of validity/dishonesty involved.
Track 1: Safe spaces create “weak people”. - So let’s say a Black American heritage group holds a gathering in some portion of a university, and designates it a safe space for black students. The conservative argument might be that there are no such spaces in society at large, so it is not properly preparing black students for the real world. This rings fairly hollow, college isn’t supposed to exactly mimic society outside college, and giving semi-private access to gathering places on campus is not unusual. Various campus clubs can usually book rooms, meeting halls etc, and while it may be a public facility, they have some right to regulate who can come in. Most of these groups won’t explicitly say “no whites allowed”, because they may not actually mind white people there (as long as they aren’t harassing/disruptive), but also because if it’s a public university allowing a group to discriminate racially might fall afoul of the law. In many examples like this, conservative media tend to exaggerate what is going on, paint a negative/distortionary picture etc, when really, it’s usually just a gathering of people who want some level of insulation from out group harassment. Most college campuses have plenty of places where white supremacists can confront black people to their heart’s content, and having some areas be temporarily managed by a student group that has rules against that sort of thing, probably is not a constitutional violation nor is it making “weak people.”
I intend to represent for both of these tracks a “bullshit” conservative example, and then an example that’s more nuanced, but in this one I really don’t have a great example of a nuance non-bullshit one that I know is real, so I will work with a hypothetical. Imagine a female law school student who says she doesn’t want to participate in moot courts with male students, because she finds men hostile, aggressive and triggering. I would actually say that isn’t something you should generally allow or want to see. There is no option to opt-out of dealing with male opposing counsel in an actual legal career, so it is a disservice to a female law student with such hangups to not be pushed to overcome them, if she can’t it will be very difficult for her to function as a lawyer in the real world. It’s also an example of where maybe you should refer someone to therapy also to overcome an individualized emotional issue that is a barrier to their specific career path.
Track 2 - Safe spaces are unconstitutional because they privatize public spaces - This argument at least has the veneer of respectability because it is generally true, in a college environment that you do not lose your First Amendment rights at the “school house door.” A student cafeteria open to all for example, would have very little legal leg to stand on if it banned students from wearing Thin Blue Line or Back the Blue merchandise. It should be noted that in many examples that right wing media highlight, no university official is actually prohibiting people from doing things like this. Instead, other students are confronting the conservatives and yelling at them. You have a first amendment right to expression, broadly, on campus, you do not have any constitutional right that immunizes you from being criticized or hollered at over your expression. However I’ll not that examples like an LGBT club in the student union bar, as mentioned above, probably would pass constitutional muster because it’s not unconstitutional to offer groups and clubs temporary private control over public assets. For example, the local public library here has meeting rooms, sometimes they are rented out to businesses, sometimes the chess club uses them etc. While they are being used, the people operating those meetings have broad rights to decide who can attend and who has to leave. As long as equal access is offered to groups, there probably isn’t an issue. So if an LGBT group is allowed to book private time in a student union bar, the only real constitutional issue is if the school would decline to allow the College Republican Club, for example, the same opportunity.
Now an example where conservatives do have an argument goes back to the University of Missouri / Mizzou protests. In that incident protesters looking to see the university administration driven out over mishandling racial issues, harassed, assaulted and intimidated a student journalist. They said that he had “no right to be there because reporters create an unsafe space for people of color protesting.” I’m sorry, but absolutely not. You are behaving in public and a reporter has an intrinsic right to report on your behavior. You have no right to assault him or knock his camera out of his hand, that honestly isn’t even controversial. The desire for safe space in that context has no compelling justification.
At the end of the day one of the core issues I think a lot of people have with looking at this issue is they assume the “other side” never has a valid claim. There are actually some valid conservative or even just civil society complaints about the specific execution of safe space policy / behavior in some contexts. Now the “weight” of discourse out there in the media and the public, is for every legitimate beef, there’s like 20,000 illegitimate Joe Rogan / Bill Maher crybaby conservative complaints. It is fairly easy to just dismiss all such complaints as the same, but like a lot of issues the right wing raises–there is usual a kernel of a real argument, usually wrapped with a lot of bullshit. Such things tend to have more power, and I think the political left undermines itself by not trying to at times push back on the kernel of legitimate truth to some of these issues.