After reinstalling, reconfiguring and even going through the trouble of enabling ssl on my webserver, I've finally found and fixed it.
I decided to watch my server logs when trying to create a new modpack. I noticed a 500 error at /api/modpack. While visiting that page myself, it turned out that database caching didn't work.
After switching back to flat file caching, everything worked right again.
Still, the fact that the technicpack.net server throws a PHP error on an API, albeit misconfigured, can't be intended behaviour, right?