Why does viewing a hdmi/displayport input on a PC require a capture card?

For that matter also is there any reason HDMI can't be run over Ethernet? Afaik ethernet has far more bandwitth than hdmi nowadays

It can, you have things called HDMI extenders but unless you know what you are doing they will not work as you expect.

HDMI 2.1 has 48Gb of bandwidth a cat5 cables maxes out at 1Gb, while a cat6 cable can do 10Gb up to 55 meters if installed correctly. Much lower then the 48Gb spec of HDMI 2.1

PC GPUs simply aren’t built with HDMI input and passthrough. The manufacturers could do it though if they wanted to.

High speed receivers are much more complex than transmitters. Just shoving some bits down the line with a bit of emphasis filter training is not that tricky. Tuning a receiving emphasis filter while simultaneously recovering the clock and performing error correction requires many more transistors, which uses more power and costs more to produce. HDMI receiver chips on digikey are 2-10x more expensive than transmitter chips.

Beyond the technical reasons, there is the fact that media companies have a vested interest in not having video input being prolific. They have a long history of fighting technologies such as VCRs. In the shadowy internet era you can let your imagination run wild with what motivations they set up in what industries to make HDMI receivers less prolific.

Also, for those talking about HDMI over Ethernet: be careful. HDMI is actually quite high bandwidth. There's a reason the cables stop working after a few feet and it isn't because they're poorly designed or manufactured (though that's surely the case for crap you get on amazon). Cat 1 million isn't magic. It's still beholden to Shannon's limit in copper twisted pair. You can't shove 25 GbE over any copper longer than a few meters of TwinAx. There are plenty of solutions on the market that are too good to be true and simply won't work at the pixel clocks they advertise, yet will happily take hundreds of your dollars and weeks of your time.

As for you last question: Search for HDMI over Ethernet.

Research NDI. It transports video with embedded audio via Ethernet. You can capture screens via software (NDI Tools is available at NDI.tv)

Check out the steambox (or raspberry pi equivolant)

I would think it's quite possible given that some systems output a video signal over USB, which is less capable than ethernet.

For your video output question, essentially the PC has a card that is capable of saying what the monitor needs to do, and ths monitor has a card to translate what the PC said into the screen output. The PC doesn't have the card that the monitor has. It isn't equipped to receive picture signals.

And kinda...technically...uh... you can run video signals over Ethernet. You just need the protocols to communicate between them, and the ports to connect them (would you sacrifice wired internet for a video signal for most desktop boxes?). Other elements are cost, compatibility, design. I've seen that some TVs have Ethernet and fibre-optic ports now.