If I were to try out ONE Wing Commander game AND ONE Privateer game, which ones are generally agreed upon as the most essential? Should I just start with WC1? Same goes for the X-Wing/Tie Fighter games.

Hard to say, Wing Commander follows the story of Christopher Blair, if you start other than with WC1 you'll miss a lot of the backstory and characters but WC1 and WC2 run on a very antiquated game engine which might put people off.
WC3 might be a good starting point too if you don't mind having a few gaps in the backstory, the manual does a good job in filling those gaps by the way.

For Privateer, the first game, the second one isn't even Wing Commander canon apart from the title, but may be worth purchasing later for the distinctive FMV sequences. Also the first Privateer has the antiquated game engine only it doesn't suffer the slowdowns like in WC1+2.

Which of the Star Wars games is also a tough question, there is little reason to play through them chronologically. I like the first X-Wing game for DOS the most as it eases you into flying the spacecraft has very good controls, shooting TIE's is fun and I think it has the best campaign overall, downside is that you need to manually backup your pilot because of an unforgiving scoringsystem. While you can continue the campaign where you was when you died but all your medals and points are gone.

Being a huge fan of both series, it's really hard to say. If you're good with old, tough games, I'd start with WC1 and play it through. But I totally recognize that I have some nostalgia there, but still play WC1 annually (if not more frequent).

WC3 is probably the earliest one that has aged the best (if that makes sense), and you'd be able to figure out plotwise what was happening. Or you could start with WC5, since it's a whole new protagonist, though I don't think the story holds up as well (but still fun!).

And Privateer is the Privateer game you should get. Great fun, here.

And for Star Wars, I'd recommend starting with TIE Fighter, sure you're playing the "bad" guy, but I think it's got the best gameplay.

No love for Descent here :(

I already played Descent 1. Pretty good but I didn't expect it to be as long as it was.

it takes time due to looking in 9 directions and moving around maze design like levels :D

I'm sure you have your answer by now but just for the heck of it:

For Wing Commander, start with Wing Commander and if you're not a fan of the old sprite-based gameplay then skip to Wing Commander 3.

For the X-Wing and TIE Fighter games, the 1995 version of TIE Fighter is the high point in the series and you're not missing out on anything by not playing X-Wing first.