Wow, it's been a while, huh?

I am looking to record gameplay from Xbox One for editing and uploading. The method I've been using is to use the Xbox "record what just happened" feature, which I've set to 1080p resolution and one-minute lengths. I then transfer them from Xbox to OneDrive, then OneDrive to PC. This works fine and all, but:
A) I need a subscription to OneDrive to have the required storage space.
B) No party chat.
C) Xbox Live storage is full (idk if this really affects anything or not, but that message is annoying).

I was thinking about using a capture card in a PC to capture gameplay with. I already have an external capture device (here) and I've used it before. It works great. My issue is that I need an entire PC to record with. In itself, that's fine, but I have a certain limitation. My dad uses this PC to do the bulk of his own stuff and this is the only PC we have running. I have some junk machines lying around and was thinking I could utilize one of those.

My thinking is that I could put a minimal install of Linux on a machine and add in the minimal software. The PC is an old one and needs a USB 3.0 PCIe card installed, which I have. The processor and memory, however, are where things get iffy. One CPU is a single core 2.1 GHz (LE-1200) or thereabouts. The other CPU is a dual core but of similar speeds (the two machines are extremely similar, but one was designed for business use and the other as a multimedia computer). In terms of RAM, I have a collective 4 GB. Hard drive storage, I got plenty (I have like 4 unused internal drives and an external).

With the setup I'm looking towards, I was thinking set up the PC as a server with SSH and then connect via terminal program/app.

My question is could I use the existing hardware and get a good recording? I tried before, but I didn't use a barebones Linux and the recording was absolute garbage. I mean, it could have also been garbage because I was trying to watch the product on the same machine. I have other hardware I could use, but they don't fit any of my existing cases. Oh, also, I have a spare graphics card I can plop in if I can unload the capturing/encoding onto it.