Thank you for your replies. It's hard to find information on these rare 3 legged beasts. Most internet searches for capacitors with 3 legs come back to HVAC posts. Having some context makes the information that is available much more logical. I was probably way overthinking their purpose, when it seems they exist simply to simplify installation and reduce assembly errors.
Those Kemet 396's are what I've been looking at but are available in very limited sizes. 1uF, 10uF, 22uF, and 47uF. And the 47uF versions are a whopping $44 each! I also need a 100uF, which is 52 week lead time, and 800pc minimum order. So clearly that isn't going to work.
Good to know I can replace these with standard 2 leg version if I use caution when installing. It also explains why one leg was nearly impossible to properly de-solder, even with my Hakko desoldering gun. If one leg is just there for support and to prevent reverse installation, it probably does not have a via (edit: solid via) through the board, just tracings top and bottom.