Ok this will probably be all out of order.
First, You have no worries about me getting upset. I live for these type of discussions on Trek. The fact that we are discussing a franchise that is over 50 years old is frankly incredible to me. Also just to be clear I see your points and even agree with some of them, DISCO isn't even my favorite Trek, I just think it gets a bad rep for having the same faults all the other shows do.
So if we go with your point of world or universe ending plots where only one person or crew could save everyone then ST3 would indeed be out. My take on your point was that all stories counted, apologies.
ST5. Considering Spock fell for Sybok, I can see Picard easily falling for him too. Picard got hooked on that video game Riker brought back, and if it wasn't for Nexus Guinan he would of stayed in the Nexus. The Nexus was kind of what Sybok was doing, removing all pain and guilt, just in a pseudo reality. Sisko I'd really need to think on it more. Losing his wife affected him in a huge way, I don't know if having that burden lifted would be something he would want, but there are moments from DS9 where I think he did.
Saru wasn't the only living Kelpian, he was just the only one there at that time.
I agree that the issue probably is the season long threat. Trek is full of planet, or galaxy ending threats, but they usually get fixed by the end of the episode Doomsday Machine, Immunity Syndrome, Year of Hell, Best of Both Worlds (really anything Borg) are some that spring to mind. But then you have DS9 which had a multi season long threat and most people enjoyed it. I think it really comes down to personal taste, and how it's executed.
My personal view is Star Trek is suppose to be a Sci Fi episodic show, not a Star Trek serialized show. TOS, and TNG held closer to that, and while DS9 is my favorite Trek, when it went the serialized route Trek went off the rails, and hasn't really gotten back on since. VOY and ENT tried to be a kind of hybrid episodic/serialized show, which had mixed results. New Trek (except Lower Decks) went full on serialized and its polarized fans. Strange New Worlds seems like it's going to follow the VOY/ENT model, so I think it will be more liked, but it's still not going to be fully embraced from the old school fans.
I don't think the first season of Disco fits in your argument of it had to be Burnham, I think you could of easily replaced her with anyone and they would of still ended the war. The actual Discovery ship though, yeah it had to be that ship, but then we wouldn't of had the show if it could of been any old ship. Season 2 did require her but that was due to the Red Angel being her mother, and Burnham was the only one that could use the suit due to her DNA. That's really no different then several other times in Trek and other shows where the only one that could save everyone would be the main character of the show.