I don't have Paramount streaming so I have only seen tos, tng, ds9, enterprise, and voyager so far. From that list here's my picks.
Captain - Picard: This was the easiest pick for me. Picard is what I see when I think of a Starfleet captain. Very intelligent, but more importantly extremely curious. He's always looking at problems with the purpose of truely understanding the deeper implications. He knows when to be decisive and when to hold back and think of a new approach.
First Officer - Riker: Picard and Riker just compliment each other too well. Riker being a little more aggressive and quick to act, but never to the degree that it could undermine Picard.
Science Officer - Spock: Also a front runner for first officer, Spocks Intellect, dedication, and problem solving skills are second to none. He's one of the most important characters in the entire cannon.
Medical Officer - Phlox: This was a difficult one to pick. In the end I went with Phlox because of his intense curiosity of the unknown and sense of adventure. He's also got some of the best beadside manners as well. His personality is extremely comforting.
Chief Engineer - Scotty: In an engineer you want someone that can push the ship in ways it wasn't even designed to do, and Scotty is the perfect example of that. There's no problem involving tech that Scotty can't solve.
Helm (I'm going to assume you meant operations officer) - Data: Another easy pick. Not only is he the best possible at this job, with an almost unlimited ability to multi-task, but he's also the best second pick for any job on the ship. He has an extreme intellect and knowledge. He does have some difficulty thinking outside the box, but if you have someone else on the ship to start a hypothesis, Data is the perfect wingman.
Navigation - Wesley: This isn't really a thing outside the original series, so I initialy thought chekov would win automatically. Then I thought Wesley could also be picked for this role, and why wouldn't you want a super genius possible next stage of human evolution Wesley.
Communications - Uhura: She's a confident officer that could take command if needed, and one of the most gifted linguists in starfleet. Not a lot of competition, but would be a strong pick even with more competition.
Tactical - Tasha Yar: Tasha has been through hell and still came out a good person. She's also the most tenacious officer in Star Trek. Unshakable is something you could say for most Star Trek characters, but Tasha would be top of that list.
Security Officer - Odo: The only time where this was a different job from tactical was DS9, so kind of a default win. Still, a very strong pick. Shapeshifting is a very useful abillity and his dedication to his job is probably unrivaled.
