So, I booted up the game and started making a massive base on one of the standard maps. After I put my brain in the right spot and started the match, nothing spawned. Not a single enemy. Not a single dropship. Nothing.
I've looked all over the forums but as far as I can tell, nobody else is having this problem.

You've probably gone over the MOID limit, aside from no enemies spawning things might break if you go over the global MOID limit (rather than skirmish defense's).
Press Ctrl+P to see the MOID count (among other things). If it's over 210 skirmish defense and some other scenarios won't spawn enemies. If it's over 255 strange things will start happening to the game.

If this doesn't help or isn't the problem, check the console using ~ and see if there are any errors there.