I would need a 25' video cable in order to have the 4K editing rig across the room share my main monitor, keyboard and mouse. Instead, I was hoping I could use 10gbe cards and vnc/rdp to get into the video editing rig from my main day-to-day pc. Would this work without issue?
For cutting and splicing of video with little regards to visual quality, a software solution like RDP etc. might work: So far my experience with RDP on 4K screens has been rather terrible even over 10GBit networks: Pretty sure the RDP codecs haven't been adapted since SVGA days and they are all CPU based.
But with Displayport and HDMI being digital and packet oriented these days, the difference between video, PCIe and Ethernet starts to blur and consequently what you want is becoming both reasonable and possible and also something that game stream-casters already seem to do quite regularly: Why should video editing be any different than playing a game and broadcasting/recording that in real-time?
Modern GPUs have hardware blocks for encoding frame buffer data at fixed latency overheads into x264 and x265 video streams way below Gbit bandwidth, but I'm afraid that those may not work at 4K or without some type of compromise on latency or quality. And you still need software to top into that: RDP/TeamViewer/VNC AFAIK don't.
What's still left is using Ethernet cabling, but perhaps without the Ethernet and certainly without the IP and software parts: There is hardware out there, which will run video signals over Ethernet Cat 5/6/7 media, perhaps even including switches (check!), but up to 100m.
ATEN is one of them:
https://www.aten.com/global/en/products/professional-audiovideo/video-extenders/ and just the first one that popped up on a Google search: There may well be more.