The old https://steamworkshopdownloader.io/ does not work anymore and http://steamworkshop.download/ only gets me outdated versions of the mods - how do I get mods from the Steam Workshop nowadays?

Probably you can't thanks to valve for shutting down everything. That's why modders should upload their mods in nexusmods or moddb There is another way that I haven't tested yet for other games but I think you also need to have the game on steam too. Here is the link and try it yourself.
https://github.com/shadoxxhd/steamworkshopdownloader

There are a small number of manual downloaders available. You have to install them on your pc. I have successfully used wokshopDL:

https://github.com/VovoloGames/WorkshopDL

It works. You do not have to own Stellaris on steam nor do you need a steam account.