Star Trek - Discovery - 3.07: Unification III

I must be miss remembering but I am sure at times Kirk introduced Spock as his First Officer.
During Kirk's captaincy, that is correct, Spock was both First Officer and Chief Science Officer. During Pike's captaincy he was just the Science Officer.
Very possibly true, but not explicated in anything shown (I guess that can also be put down to poor writing or the cutting room floor).

If you look at early episodes of Discovery, several other Commanders and Lt. Commanders are mentioned, but some of these were killed on-screen and others haven't appeared or been mentioned since Season 2. We also know 50 people stayed behind with the Enterprise so a logical extrapolation is some of the other higher-ranking officers would have stayed as well.

Although that whole 'We want to be kept together as a single unit' thing shouldn't have prevented them getting new crew to make a full compliment, even if they resisted (though why resist?) Of course, Starfleet itself may be very shorthanded right now. Very probably it is.

That is a good point and they do have Adira on board as a new recruit, so presumably could have taken on more. Manpower may be part of that, but I have a hard time believing Starfleet couldn't spare 2-3 people just to help the Discovery crew acclimatize to their new tech. Of course, maybe they have and they just haven't appeared yet.

Did he though? There is a period in which we don't see him, in which he could have been promoted through intermediate ranks, although I agree that promotion would still have been fast..

We see Sulu as helmsman in The Wrath of Khan, with the plan of being promoted to Captain and being given his own ship on the completion of the training mission, so it would appear that there wasn't an intermediary position. Of course, the whole stealing-the-Enterprise thing got in the way of his career, but we do see him still as helmsman in The Final Frontier and then, three years later in The Undiscovered Country, being Captain of the Excelsior and noting he'd been in that position for three years.

Wasn't "Number One" the Majel Barrett character, now to be played by Rebecca Romijn (if they still make a Strange New Worlds series) just "Number One" and nothing else, with Spock as the Science Officer on Pike's Enterprise. And one of my points is that Tilly doesn't seem equipped or trained for that HR role either, though she can grow into it, of course.

No, Number One was a helmsman as well (which of course undermines the Sulu/Owosekun argument!).

Anyway, thanks for that well argued reply to my question (y)

No problem!
I've been thinking more on why Tilly wasn't given a promotion to Lieutenant, even if it was field promotion during last season's events, but then I remembered poor Harry Kim who spent a decade as an Ensign on Voyager (including the year of hell and other skips in time).