I mean, yeah, that's a very healthy looking kitty. I get how she's emotional. And I know the way she's holding the cat isn't painful per say, but long term holding the cat like that can't be too comfy.
Picking them up like that and then readjusting once you picked them up is totally fine. But she seems like she's just walking home for 20 minutes like that. The cat is very calm, so she's clearly ok with being brought home, so at least she's getting a new home.