Yeah, agreed. It literally feels like a personnel problem. Not least because the 24th Century Enterprise D, admittedly with a much larger crew, did have a ship's counselor, and Discovery is a 31st C. ship now ... with a crew that was selected in the 23rd, minus the enlisted folks who sensibly bailed to the Enterprise (no-a-b-c-or-d) before Disco went through the wormhole. (Which, in retrospect, makes it a bit impressive they survived as long as they did before getting the handy smart-matter workings and DOTs.) Like, I know Discovery is a much smaller crew than Enterprise D, and Culber's job of keeping people alive is a little simpler with all that fancy new tech, but there nonetheless seems to be a workload that's not really being efficiently distributed.
That personnel problem kinda seems to be the source of the problem, too, since it all relates back to this time-displaced crew. Everyone's coping with losing everyone they've ever known and having no personal attachments outside Discovery by putting themselves into their work, no one wanted to leave the ship or see it under new command last season - but this isn't Voyager, nobody's do-or-die trapped there, and increasingly there's a whole Federation out there to fill in the holes. And contrary to the crisis-mode response of giving the crew something to latch onto last season, the three people who've left the crew for a while to connect with something else familiar and pursue a goal outside Discovery (Saru, Tilly, and Nhan) seem to be the most well-adjusted and comfortable in the new present of anyone and more or less living their best lives.
It doesn't seem beyond reason to transfer on another nurse practitioner or something. And maybe encourage some of the crew who aren't important to the season arc to take a position on not the only ship in the galaxy with an instant teleport drive that's on indefinite save-the-universe duty, take up a hobby, and get some perspective on work-life balance.