Whenever I try to launch the daily run, I get an error message telling me that I have to be online (even though I'm connected both via lan and wifi)

Same here

If you're using Linux, it won't work on that. I spent quite some time trying to fix or work around it some time ago, but I wasn't able to. I did hack around things to get the awards for completing them by hex editing some files, but I forgot what I did exactly (it's been a while).

I wouldn't expect any fixes; it's been broken for years.

Or maybe it's something else; it seems the game will just throw the same message no matter what error occurs.

It does work, but only sometimes. No idea what it depends on, though.