Okay admittedly I have stopped watching in season 3, but in my opinion what sums him up is 1) neurotic 2) ridiculously occupied with his emotions and personal life.
Completely unprofessional. Of course he gets by just fine because he is also the smartest guy in the universe and is irreplaceable to the operation of the ship, but if he were held to normal standards he would probably have lost his commission twice over.
Honestly I agree but I also give him the credit that Starfleet was never something he wanted to be a part of in the first place. I LOVE the character and would change nothing about him but he's not Captain material. It sort of makes sense he doesn't have any of those qualities of a Captain when he was forced into service in the first place and really not that long ago. He was pulled in during the Klingon war and the shows events happen either during that or not long after so he's still pretty fresh to the fleet overall.
Completely right about the unprofessionalness too. That seemed to be allowed in Starfleet moreso in their early history, looking at you Bones and Scotty, but even Stamets pushes that at times. Chain of Command is more of a loose organizational structure to him than a true chain of command and that does hurt me a little personally. I feel like he'd probably be able to put the good of the ship before himself (We did see him sacrifice himself with the hundred-odd microjumps in Season 1) if absolutely needed but also that he'd never want to be in that position in the first place.
Yeah, my first thought was a clear lack of stoicism. Accepting the hard things and enduring forward fast because a Captain must. Everyone is depending on that. I can imagine this guy trying his hardest and then just breaks; blanks out, freezes up, is overwhelmed by what has happened and what is happening, instead of only thinking about what's coming next, which is what Captain's have to do and actually do.
The logical Captain performs well from remaining cool. The emotional Captain performs well by being selfless above all. He is clearly neither if something is personally impacting him.
To simplify, this guy would be good if he were more Vulcan.
The character would make a pretty lousy Captain. Regular Captain anyway. Temporary in a crisis or something? Probably passable and good but Stamets hasn't shown himself to be particularly strong when it comes to that aspect of his role. He's very impacted by emotions and focuses on that and his science. Then again he never wanted to sign up with Starfleet in the first place. He got roped into it and has only been part of it for a handful of years, honestly not much more than Tilly. Starfleet roped him into the Klingon War with his research on the Mycelial Network. That only started a year before Episode 3 of DSC. Meaning that Stamets had been part of Starfleet for, at most, a year by the time we first see him. Explains a lot as to why he's so angry and bitter. Also why he changes so much on that front throughout the show.
I would say this is pedantic but given I named my character after him...
Stamets isn't actually an engineer! He's from Science. He just has a lab set up in Engineering Test Bay Alpha. We've never actually seen the Chief Engineer on Discovery and have absolutely no idea who they are. There was something incorrectly displayed on screen in Season 1 but the producers and such admitted it was a mistake. That mistake did say that Stamets was Chief Engineer but there were a few other things around it that pointed to it not making sense.
The only Engineering characters we've really seen regularly is Commander Jett Reno and Tilly. Kinda weird that. Everyone else is either command or operations.
I think Jett Reno is supposed to be the chief commander of the USS Discovery, considering she was the chief ___ on the USS Hiawatha. Though the show never explicitly states it, I think it has to do with very old Trek ideas of no enlisted personnel. ie, everyone is an officer who's gone to Starfleet Academy.
The only Engineering characters we’ve really seen regularly is Commander Jett Reno and Tilly. Kinda weird that. Everyone else is either command or operations.
I think you may have explained why Disco never felt right as a Trek series to me: Not enough engineers!