The "crying" complaint is questionable because its apparent premise is that the crying is somehow happening inappropriately or at unprofessional times and that just isn't the case. People trying to pretend otherwise often seem not to actually know anything about the context of how events happen on the show. It often sounds like an excuse for some other issue that people aren't willing to actually elucidate, like a mere pretext; much like the "whispering" complaint that preceded it in fashionability. (Anyone remember when Burnham was supposedly "always whispering"? Good times.)
For example, I remember complaints about their having some kind of "kumbaya crying circle mid-mission" in one of the episodes of Season Four: and that's literally the episode and the scene where the crew learn that Species 10-C communicates through emotion-bearing chemical compounds. That salient fact somehow never makes its way into the complaints, and way too much of this stuff is like that: examine the actual context and the complaints turn out to be petty, trivial, laboriously illogical.
Factually, in-context, the DSC crew sucks it up, deals with crises in the moment and deals with their emotions and mental health later just as you'd expect any other crew to do. Burnham is no different and is perfectly acceptable for pulp SF values of inspiring leadership, even in early seasons where she's rough around the edges.
Where DSC is different from other Trek shows is that it makes an explicit decision to really show the "dealing with emotions and mental health" step, and to show it as a consistent process that is part of the necessity of the crew's survival. In this sense, it extends ideas first seen in TOS -- where McCoy served as Kirk's confessor -- and then in TNG where we see Troi play a similar role in talk therapy sessions with a broader range of the crew. DSC's treatment of mental health is more extensive than either, the range of permissible emotion much broader, the people permitted to engage in it skewing far more female and of-color and gay and nonbinary, the entire definition of a hero much less based on masculine stoicism.
That's a noticeable difference and an interesting one. And (bracketing out the issue of misogynoir regarding Burnham specifically) I think it's the real root of the "crying" complaints. The basic model of original Trek was being idealized Navy. The depiction of heroism in DSC is, by design and particularly when we get to the 31st century, un-military. The degree of emotional unguardedness Burnham and her crew have with each other, no matter its timing, would get you chewed up in any IRL military and reflexively, that's the formula a lot of Trek fans (consciously or not) expect shows to follow.
I understand that frustration at some level. A different version of me, a younger version of me, might have become a DSC "hater" out of this same instinct. Fortunately it hit for me at a time when I was ready for something different, when I was putting these kinds of things in perspective, and so I was able to enjoy it.
Ultimately, I think it's perhaps the show's single most interesting decision. I think it's a key part of what makes DSC stand apart from other shows. I also think it spoke to a cultural moment -- the best Trek tends to speak directly to its cultural moment rather than nerd obsessions with "continuity" and "verisimilitude" -- where our society needed (and needs) to step back and carefully examine what it thinks "heroism" is and why. I think that will be an enduring part of its legacy, and it's what earns Burnham her place at the Captains' table. I wouldn't have it any other way.
