I have it connected from the Xbox One S straight to the PC (without a TV), but all it shows is a black screen. The Xbox works perfectly fine on the TV. Is it completely necessary to have a TV to use the capture card or is it something else?

Did you disable HDCP? This is a technique to protect copying protected content like netflix and DVD movies, but shouldn't protect games. You can disable it somewhere in the settings of your Xbox One, use google if you can't find it yourself.

Are you using video capture device and selecting "Game capture HD60" ?

I also had to run the elgato hd software first before OBS or SLOBS would recognize the source, so try that. Also the elgato software while not at obs level is still very nice.