Clearfell, I don't really understand what does and does not work. Please correct me if I'm wrong:
- Your PVR is broken
- You have a satellite dish as an input (?)
- Your TV has a satellite tuner as well (?)
- You want to record TV with your computer
Your PC has HDMI ports but I guess those are output ports. However you need the TV signal to get into your computer first to be able to get it out to a TV and/or monitor.
Some time ago I have used a Hauppauge TV capture card and I was content with it. They now have a whole list of possible solutions. Back in the day it was important to have hardware compression, with current hardware this might not be that important anymore - but for robustness I would still prefer it. They even have their hardware output IR signals to steer a set-top box
Check out Hauppauge's product page:
https://www.hauppauge.com/pages/products/prods.htm
Only downside is that their hardware is supported by Windows-only software, however they DO have a Linux support page you might find interesting:
https://hauppauge.com/pages/support/support_linux.html.