Yep, the Video Source plugin won't do randoms, or play with a delay timer.
Problem is, you can't chromakey the VS plugin on Classic, and the version on Studio doesn't appear to support playlists; just one video file per source, even if that one video could be looped.
If you can export to AVI-with-alpha, OBS Classic should be able to handle the alpha transparency natively for things like fades or slide-in/outs. At least that's what I'm told by someone who used it, I don't myself as I have yet to learn AE. Would recommend slapping together a test item first to make sure the VS plugin does keep alpha.
You could then have a reusable 'spacer' video at that point that's just a single frame of alpha the same size as your template lower third, and insert it into the playlist however many times to allow a variable delay between the content videos. Also shouldn't be too big with essentially a single frame and a long duration.
After that, just set up a long playlist in the Video Source plugin and set it to loop, and make sure it's a Global (otherwise it would reset each time you switched scenes to the first video on the list).
Alternately, you might be able to embed the videos via a local HTML file of some sort and do flow control with CSS or javascript, and run that through the CLR Browser plugin. But that one gets more involved.
Short version, no, there's no turnkey solution for that.
But there are ways it could be worked around creatively.