Don't ragequit & uninstall the game! Captain difficulty is doable.
Remember to save often in the battle.

Save your command points for this battle. I succeded by speeding the diplomats in top border, you must ALMOST rush. Test and try the good rythm. you must have offensive ships.

It is said you can succeed too by turtuling instead, in left corner, destroying all the many enemies with defensive ships.

Have you played farther?

I've saved the diplomats already, it was decently hard but very manageable with one or two retries, even without using up many commander points.

I'm stuck at a mission a bit farther though, where you have to "slingshot" around a moon to get away (don't wanna spoil more). I've tried it 10 times at least, there's just a million enemies, they spawn faster than I can kill them AND you have to get to the other side after killing the objective, so you actually have to go THROUGH them all. It seems pretty much impossible with my chosen upgrades and I'm considering restarting the game to make upgrades more fitting for that battle.

The issue is that I can either move, OR shoot, and if I move it's almost guaranteed that one of my units dies the next turn because there's just SO MANY enemies on screen from the start that they can almost oneshot a unit every turn.

I'll probably just restart tbh, I really enjoy the game so it's fine. But maybe there's a trick?

I'm also stuck at the same mission, and I suspect that I may have fucked myself with my upgrade choices earlier as well. I think I need more shield upgrades to beat it.

Scenarios are badly done, not enough tested (Captain so difficulty is the normal one). That's sad that is said (didn't play yet) in next game instead of balancing they made it too easy and we can change the sets of the ships freely.

Well, of course missions get easier the further you progress. This is not only due to bad balancing but due to the player, who improves and figures out the game by developing the right strategies. Yes, on Captain difficulty the game is hard, but only when you play it the first time. On a second playthrough captain is easier, still not too easy. But what is a game, when it's to easy anway.

And by the way, you can always change the difficulty, its no shame to reduce the difficulty if you cannot win a certain mission. Of course, the missions, which poses as an end fight of a particular chapter are often harder than sidemissions ;) There is one sidemission though, which is very hard and considered the hardest mission in the game. It is not mandatory and I reduced the difficulty on my first playthrough there as well :)

There are some guides for purchasing the right upgrades. I think soon such a guide will also make an appearance in this forum :)

My ryder's don't show up in this mission, which appears to make it impossible. I thought it was intentional, but after looking at some online videos / walkthrus, it seems to be a bug.

Has anyone else encountered this? What can be done?

I haven't encountered the issue, but the game does glitch every now and then. Usually it gets fixed on a retry.
Perhaps try loading a save, before the battle, and see if that fixes it.

I eventually did find mention of the issue in a steam thread. Basically the simulation mode is glitched; if you lose your ryders and exit the simulation they're gone for good (at least in some circumstances).