The good news is that 4 years after the release of the Switch and the problem first appearing, somebody finally found out what is actually wrong, and how to permanently fix it. The problem is a lack of pressure on the underside of the thumbstick. And the fix is taking a small square piece of thick paper (e.g. from a business card) and placing it inside the joy-con, at the back of the thumbstick. And it works! The drift of my joy-con is completely gone!

I'd recommend following the steps of the IGN video on the fix. The only bad news is that unless your tool box is unusually well equipped, you probably will need to buy a tri-point Y00 screwdriver, and maybe even a Phillips #0 screwdriver (I had the latter, but the former is a rather specialized tool). Fixing your joy-con will void its warranty, but I'd rather have a working controller without warranty.

I'm not claiming any discovery here, I just discovered the fix on YouTube, with the original video explaining the fix having 1 million views. The IGN video is just a bit more structured and easier to understand for people who aren't in the habit of disassembling electronic devices. Basically, if I can apply this fix, so can you. :)