End of 10 is a campaign to move people over to Linux with Windows 10 support ending
(www.gamingonlinux.com)
I think you are right about the Elgato PCI-e models not working on Linux unfortunately.
In case you are ever in the market for a new capture card in the future:
99.9% of USB capture cards work out of the box. Alternatively, if it has to be PCI-e: Blackmagic provides first-class Linux drivers for their PCI-e cards.
We use some Blackmagic cards for larger projects and for the smaller ones we use Elgato USB capture cards.