Which set-top box?

The price should be under 170 euro.

It would be good if everything works out. (YT, Netflix, Spotify etc.)

I can stream my phone and PC screen on it

Cable or Sat?

More for online services. Cable Is the TV still receiving?

You can easily do something. What you have in mind goes well with googles crome cast as well as with amazon fire tv. The price would even include an apple tv. That can be worthwhile, but only if you otherwise have apple devices.

Otherwise my favorite would be the H96 Maxx Internet Box based on Android. There's a set with a wireless keyboard for well under 100 euro