In principle, it is possible to use this repeater to integrate a network device that only has LAN (but no WLAN) into a WLAN network - I myself have operated a non-WLAN-capable network printer in this way.
However, I cannot judge whether this also works in the case described and depends, among other things, on the quality of the WLAN connection; a set-top box needs sufficient bandwidth for television to work.

Yes, this is possible as long as the M Budget router has Wi-Fi and the radio signal reaches the other room.