bold of you to assume your hdmi cable won't naturally form some hoops.
I don't think anyone is currently making an all-purpose VR headset which can also do HDMI mirroring though.
Technically you can just remote desktop into a computer from your VR headset, but that won't work in every scenario.
Maybe you could convert the HDMI input into MIPI and feed that into the SoCs camera interface, but I think headsets like the Quest are already pretty much maxing out the SoCs camera capabilities just to read in all the actual cameras used for inside-out tracking. There's no bandwidth left to shove an extra HD video feed in as well.
tl;dr HDMI input that turns the device into a dumb PCVR headset: easy. HDMI input that mirrors arbitrary video: hard.
The second part needs some explaining. One undocumented feature of Quest 3 is that it supports (some) USB-C Ethernet adapters. There isn't really any UI for it that I know of; things just work so long as DHCP is there. This then gives you a direct wired 1GBps link to the PC, which Steam Link will happily utilize.
XREAL as well as some drone FPV goggles support non-/partially-head-tracked HDMI input, albeit with much smaller FOV for comfort reasons.
https://goovis.net/products/g3max
(This particular one uses USB-C for video input, but they also sell an HDMI adapter for it.)
Of course, in practice it's just a display and can be used for any purpose. I do appreciate the fact that you don't have to mess around with all the usual VR setup chores with these - it's really very plug and play.
(xreal is also in this category imo but at least its a bit cheaper)
And yeah, $1K is quite a lot, but the 65 degree FOV and 2560x1440 (per eye) OLED that you get for it really does look amazing - I haven't regretted that purchase in the slightest. It's rather bulky, though, although still nowhere near as heavy as VR headsets. Still, not something I'd want to carry on the go.
I hope we'll get 4K per eye some day with this tech. Whichever brand does it first for reasonable $$$, it'll completely replace my primary display.
PS: I have a barely used Meta Quest Pro for sale. ;)