I don't believe MO can do anything to make most grad students comfortable posting here. It wouldn't be compatible with the culture of the site or how its users see it (and some of the comments on this topic and in the linked posts make this point rather blatantly).
Speaking as someone who has used MO intermittently almost since its inception but fundamentally feels uncomfortable here and always exhausts other known options before daring to ask anything, what has always driven me away is the sense of gatekeeping. (This isn't substantially better now that I have positions and publications under my belt.)
Answers typically get far more upvotes than questions, leaving the impression "Great answer to a stupid question" for most things asked.
Many times in the past, I have seen comments and answers that imply that the question is obvious or beneath them.
One gets the sense "Some of the world's top mathematicians are listening; don't waste their time." (I actually thought I was paraphrasing this from the site's guidance, but it seems that's something half-remembered from an answer.)
I understand that one sees MSE, it looks like a cesspool of homework help, and one wants to do anything to avoid that, to keep the quality of MO up, as it were. Great. But the cost of keeping a high bar to entry is, expectedly enough, that it discourages entry. I would venture to guess most mathematicians, and surely a greater proportion of students, see their questions, and possibly themselves as mathematicians, as too low-caliber for the site.
(I don't think it's a coincidence the most popular posts on the site are the "big-list" questions from the early days that engaged a broader swath of mathematicians, but it was decided somewhere along the line that that was not what the site's more dominant users wanted it to be.)
This is a reputational and cultural issue that I don't believe is "fixable" now, if indeed anyone would want to. Its[Edit: It seems to me that i]ts regular users wanted MO to become an elite institution and got their wish. It's no surprise that now even "the right sorts of people" are often too cowed to post.