As I understand it, the only output from the ATEM TVS is SDI correct?
No analog or component etc?
Nothing seems to want to talk with the Black Magic stuff. I know it's cheap, but if it does't work with anything, what good is it?!
The TVS has both SDI and HDMI outputs. The first is the industry standard for professional digital video and the second is something of a "pro-sumer" standard, as well as a common interface for monitors and projectors. So the TVS does in fact work just fine with an overwhelming majority of digital equipment out there.
If your application requires analog outputs, perhaps an analog switcher would be a better fit than the TVS, which is fundamentally digital.