This is a really cool way to think about anti-villains and makes me want to revisit some of the villains from my favorite shows to see if they fit into this role. Off the top of my head the only anti-villain i can really think of is deadpool just because he's such an iconic anti-villain but now i'm going to have to rethink a lot of villains.
In terms of sympathetic villains though I really found the tv series Gotham to be amazing for this reason. The backstories to characters like the penguin, riddler and mr freeze are amazing and truly make you both sympathize with them and view them in an entire different light.
I absolutely love villains. If a villain is well written, chances are they'll be my favorite character. They're also the most fun to write! :)
The interesting thing about Deadpool is that he falls into both the anti-villain and anti-hero categories, since his allegiances shift and change so often during the course of his story.
Do rethink a few of the villains that you may not have thought too much about before. There are many of sympathetic or anti-villains out there, but they tend to get dismissed by most people. Mostly just because most people watch or read something with a "surface level" interpretation. Not that everything needs to be overanalyzed, of course, and sometimes a villain is just a villain, but it's a lot of fun to dig a little deeper. At least it is for me. :)