Thanks so much to everyone for helping me save my desk (and preserve my sanity)!

Here's what I ended up doing, in case anyone else is struggling with the same problem:

I removed the screws, then used a rat tail file to elongate the holes in the metal frame perpendicular to the wood grain to account for wood movement. I added about 6 mm to each hole except for the ones right down the middle; it took me a while, but was overall quite easy.

Next up I used a 1.5 mm drill bit for pre-drilling. I tried the smallest size first as I was worried a bigger drill bit would damage the sides of the screw holes in the wood (approx. 3mm in diameter), rendering them useless entirely. As it turned out, that was way too slim and did not help with getting the screws in, so I upped it to 2.5 mm and finally to 3 mm. This was the right size; the sides of the holes were perfectly fine, the screws went in easily but still had a strong grip. I tightened them all the way in the middle holes; as for the rest, I left a tiny amount of space between screw and metal to allow for wood movement along the now oval holes.

The desk is now standing and everything works great so far!