Okay: So I'm running Zoom v5.1.2 on El Capitan (10.11.6), and Zoom says my version can't use the Virtual Background feature in Zoom. Is there any patch, work-around, or some other way of enabling the Virtual background with my software?
Yes. There are minumum hardware requirements for that, mostly the CPU. You need at least a dual-core i5 processor, or higher.
If you want to use the 2009 mini that you have asked about, that has a core 2 duo CPU -- not enough power for virtual background, unfortunately.
more info - https://support.zoom.us/hc/en-us/articles/360043484511
(even with an i5 dual core i5, you likely need to be running 10.14 or higher)