Bought games don't show up in library on new PC

I’ve got some problems with my steam library.

After buying a new pc, I logged into my steam account there and downloaded a few of my games I was in the mood of playing, including some free games and one of my paid games (Game Dev Tycoon). This was a few months ago. When I wanted to download the others, they weren’t in my library, the free games (e.g. TF2 or Duel links), as well as the paid games (e.g. Mega Man X Legacy Collection, Portal 1+2 and Table Top Simulator).
When I searched for them in the Steam store, I could simply download the free ones, but there is a price shown for the paid games.

In the Steam forums and StackExchange, I found two tips I already checked: Being logged into a wrong account or having the wrong filters.
I checked whether I was in the right account by changing my avatar on one pc and checking whether it changed on the other one, which was the case. Also, as I mentioned before, I still have one of the games I bought. My account info also aligns.
On my old pc, when I use the standard filters with tools hidden, steam shows me that 19/114 objects are shown (all of my games minus the tools). With the same filters on my new pc, it only shows that 8/104 objects are shown. If I disable said filter, the hidden tools, which I didn’t ever download, btw, are shown, but no games other than those that were shown with the filter.

Do you guys have an idea? It doesn’t help that I can’t really contact Steam support anymore and their self-help guide doesn’t really work in my browser