I use StarTech's USB3HDCAP device, and I output from an old broadcast monitor (Sony BVM -- it's awesome) that my husband and I use when playing our old consoles (PS2 and older). I think, in this case, we also split game audio between our speakers and our StarTech device, capturing audio separately from video. If we need to capture from a modern console, we use an HDMI splitter, to send the signal to both our TV and our StarTech. The StarTech connects to our PCs via USB 3, where we use OBS to stream or record.

Modern consoles are much easier to capture from, since most capture devices are meant for them, seems. Capturing video from older consoles is simpler than I described if you aren't picky about video quality, but it can become tricky otherwise.