Relative to this discussion though, do you then "flatpick" with the thumpick. Meaning, do you pick down and up with the thumb pick?

No. It would be too awkward to do this with a thumbpick as the edges aren't bevelled enough. For those instances to do what you originally mentioned, I'd use a flat pick, and the three free fingers to claw the string chords....or, I'd use the thumb pick and a fingerpick on the index to do a 2-fingered style of flat picking, with the 3 remaining fingers to pluck chords. I'm pretty flexible. :-)