To take the load off the gaming PC you need a capture device connected to the Mac to take the video feed from the PC, so all the PC is doing is playing the game.
Streaming RTMP to the Mac and then streaming more RTMP to Twitch from the Mac doesn't actually make things any easier for the PC. The PC would still have to encode the stream and send the data over the network. It doesn't make any difference to the PC whether it is sending it to Twitch or across the room.
That being said, if you really want to try this, look for Red5. It's a custom RTMP server; you can set up OBS on your PC to stream to a Red5 server running on your Mac, and then load that in a web page and capture it as a source in OBS on the Mac, and then stream that to Twitch.
But again... you still have to have OBS running on the PC, encoding your video, and sending it over the network to the Mac... so you might as well just stream directly to Twitch.
If you do decide to buy a capture device for use with the Mac, go with something from Magewell, BlackMagic, or AJA. Which one you want will depend on the resolution and framerates you intend to stream at.
Under NO circumstances go with Elgato. Their hardware on MacOS ONLY works with their software, not OBS-- they do not provide a standard video driver. This forum and the OBS subreddit is full of Mac users who bought Elgato devices because they claimed to work with OBS and on Mac-- without realizing it doesn't do both at the same time. It's OBS on WIndows, or Elgato software only on Mac. The only workaround is to run the Elgato software and then do a window capture on it, and the performance is terrible.