Can I stream games over Cat6 with Sunshine + Moonlight?

I’m planning to build a gaming PC/server combo in a cupboard and use my existing Cat6 cabling to stream games into my bedroom.

Here’s the setup I have in mind:

• PC/server in the cupboard running Sunshine (output).
• Cat6 cable from the PC into the router for internet + LAN.
• Router also feeds the Cat6 run that goes to my bedroom wall jack.
• In the bedroom, I’d patch that wall jack into a small Ethernet switch, then run cables to my TV and monitor.
• On those devices I’d run Moonlight (input) to receive the stream.

My understanding is:

• Sunshine captures the game and streams it out.
• Moonlight receives the stream and sends controller/keyboard input back.
• Cat6 can carry both internet and the video stream at the same time, so I don’t need a separate HDMI run.
• Latency over wired Cat6 should be low enough to feel like a direct connection.

Does this all sound correct? Has anyone here done a similar cupboard‑server setup with Moonlight over Cat6? I want to be sure I won’t be disappointed before I commit to building it.

TL;DR: Can I stream games from a cupboard PC over Cat6 using Sunshine + Moonlight, with router → wall jack → bedroom switch → TV/monitor, and expect low‑latency performance?