So I think this must be 99% the inexperience of the people at Planned Parenthood. (It's not necessarily a bad thing that they're inexperienced, I don't expect PP to have the best of the best or anything. I expect them to have someone who can handle the average patient, and they do. They aim to make their care affordable, and so I don't expect them to have super experienced phlebotomists if most patients will be fine with who they have now.)
I went to LabCorp this morning, she found a vein, she stuck it, simple as that. I didn't feel any bit of faintness, It went SUPER quickly, she got plenty of blood. I'm not even sure if she tried a smaller needle. I am definitely getting my future bloodwork done there.