@raccoon Guessing you are using a laptop rather than a desktop if you say you don't have a monitor already?
My pennies' worth here is consider searching for a 'KVM switch' as well as 'capture card'. They're functionally the same hardware, but KVM switches might be better optimized for low latency than a typical capture card. KVM switches are expensive, but the advantage of being 'office' equipment is that you might get a second-hand bargain more easily.
@raccoon Sort of - the basic KVM switch is just connecting two or more computers to the same peripherals like you say.
A networked model ("KVM-over-IP") takes this one step further. The machine (let's say your games console) is plugged into it and "sees" a monitor attached to itself. The KVM-over-IP unit doesn't have an actual screen, but instead sends the video signal from the console to another computer on the same network. You'd see a window on your laptop with the entire video output of the console inside it. You could resize or hide that window, and the console would still happily think it's got a screen attached to it.
If you can connect a gamepad directly to the console that would work perfectly, but sending the gamepad signal through the KVM switch (because you want the console to live in different room to your laptop, for instance) is trickier.
I'm sort of hesitant to recommend it too much really, just because of how expensive these devices are. But if you could pick up one cheaply, I think it would make for a very pleasant experience. Remember Google Stadia? It'd be like your own personal version of that :)
@raccoon Oh, and be careful to make sure it's actually a KVM-over-IP unit and not just a KVM switch with a network switch built-in... that's probably what the cheap ones you're seeing are, although I'd be delighted if I were proved wrong!
@raccoon Yeah, I might have introduced you to a red herring there, sorry... IP is definitely not the bottleneck; single-digit-millisecond latencies are possible (I've achieved them in certain setups) including with high-resolution video. But I can't actually find any low-latency KVM-over-IP devices for sale 
I guess something that is possible does not equal something which is purchasable 
Hello again @raccoon ! Sorry for the delay in getting back to you - I wanted to give a proper response but that also meant replying to you went in my stupidly long backlog of F-Droid tasks 
I'm so glad you find F-Droid valuable. It always makes my day to hear how people appreciate using it 
Regarding the Google changes, we're actually having a special meeting this week to discuss that. We've already spent hours discussing this, so unfortunately even the threat of Google introducing these new restrictions has already been taking valuable resources away from improving F-Droid :(
One thing we are doing is trying to improve the relationships between F-Droid and other free builds of the Android Open Source Project - things like GrapheneOS, LineageOS, CalyxOS and so on. We already shared a stand with e Foundation (of eOS) at the last FOSDEM conference, which was really great! The reason why this is important is that these operating systems won't be subject to Google's restrictions, so users will continue to be able to use F-Droid as normal.
For the users who will be using Google builds of Android, which unfortunately will still be the majority for a long time, we're exploring some options to make it easy for users to run apps inside virtual machines. This still won't stop Google having ultimate control, but will take a lot of the hassle away from using F-Droid after the new restrictions come into effect, and it should prevent Google selectively banning specific Free and Open Source apps.
Also a big shout-out to @IzzyOnDroidOrg who have been doing amazing work as well. Their efforts have been absolutely invaluable in getting the [https://KeepAndroidOpen.org] campaign off the ground (F-Droid is one of the 71 organizations backing it.)
Thanks again for asking and I would be more than happy to answer any questions you might have or clarify anything I was unclear about 
@raccoon We really appreciate the cheers from users like you
Don't feel obliged to donate or anything :)
GrapheneOS hasn't been super positive about adding F-Droid by default in the past, but that's exactly the sort of thing which hopefully we can establish a rapport with them for, and see if we can resolve any of the objections they have :)