I too have this use case troubleshooting headless computers around the house. I saw an ad for https://www.aurga.com and bought one for $84. It connects to an app on my Android phone as display, keyboard, and mouse. Minor discussion 7 months ago: https://news.ycombinator.com/item?id=38609526
Long ago I bought a mini keyboard + mouse combo for input; the custom wireless USB dongle edition I have (strongly preferred over Bluetooth when troubleshooting) is no longer for sale. https://amzn.com/dp/B00I5SW8MC
This looks quite interesting, thanks for the link. It does seem to require a native viewer instead of having a web interface. I would really prefer just a website like the PiKVM. Might still get it.
I have to do an off-topic rant though. The marketing page you linked to does not really state what this device does. It has a nice look into the case and a lot of buzzwords, but nothing like a small section with "HDMI Input" or "USB keyboard emulation". Even the shop page is somewhat light on details, but it at least shows (in GIFs only) that it works as a display and has a USB port. If I wasn't given your comment as context, I would likely not have gotten the use case and closed the tab. Based on the form factor being similar to a Fire TV stick etc. I would have assumed you plug it into a Hotel TV or similar to work on that.
EDIT: Saw your edit now and I think it is kind of funny that the old HN thread is also mentioning the marketing.
If you don't want to spend a lot of money on this:
1) Get a Bluetooth adapter.
2) One of those cheap Chinese keyboard/mouse combinations like the one you linked (the are many more).
3) Get a cheap Chinese portable monitor with HDMI/USB-C.
4) Optionally, a powerbank for the monitor. Or get one with a battery.
You'll be done with a similar price as this 'Aurga'. Back in the Nokia days, you could use Nokia Linux device as a remote keyboard/monitor/mouse, too. So e.g. Nokia N800/Nokia N810 a better cyberdeck than a Sharp Zaurus.
I have that keyboard and gifted one to my parents. Our use case is the odd chance you need to input text or use a browser on a Smart TV. Works so much faster than the on screen keyboard. With many Smart TVs just being Android under the hood, it just works.
I find for server troubleshooting, I usually have no problem grabbing a random USB keyboard. The bigger problem is finding a screen at a convenient location and connecting that one. It often was easier to carry my server to the screen instead of the other way.
On the topic of niche input methods, I also have an "air mouse" [1] with a full keyboard on the back for my Kodi system or when connecting my desktop to the TV. I essentially never need to use it, but it has come in handy.
I have one of these and it's great for this sort of thing. The whole keyboard runs off one AA battery which lasts forever, and it even has a storage spot for the USB dongle.
Those combo devices are incredibly useful in a pinch -- I keep one in my backpack. I've used it countless times while working on interactive installations and also while messing around with Pis at home.
Long ago I bought a mini keyboard + mouse combo for input; the custom wireless USB dongle edition I have (strongly preferred over Bluetooth when troubleshooting) is no longer for sale. https://amzn.com/dp/B00I5SW8MC