I experience the same thing. I have used a Micomsoft sc-512n1-l/dvi using the pass-thru and confirmed that the pass-thru signal was uninterrupted while the black screen appeared. I suspected the card may be defective so I purchased a Startech USB3HDCap (same as OP). I wasn’t able to get the card to work at all on my desktop, but I tried it on my gaming laptop and experience the same problem.
I have tried every version of the Micomsoft drivers I could find – it seemed like it occurred more often with some than others but the sporadic nature of the problem makes it hard to tell.
As with OP, the red light on the OSSC isn’t coming on, I am also using a modded SNES mini and same OSSC firmware.
I have only been using my SNES lately so I’m not sure if it’s happening with other consoles….it’s hard to test since it’s so sporadic but I will post an update if I see it.
Additionally, the screen isn’t always black when it fails – sometimes it is green. Here’s an animation showing the issue: https://i.imgur.com/v0K2jGj.mp4
Edit: also – as with OP I am using OSSC 1.6