I can't speak for NJ, but I trained as a phlebotomist in FL because I got a blood clot from a bad blood collection (actually donating blood) and wanted to know more about it.
It was an 8 week class taught at a hospital. The very first week, we started "sticking" each other on Friday and by the following Thursday or Friday, we were out on the floor sticking patients, wearing scrubs color-coded for the "lab".
We hadn't even passed the class yet... We also did EKG's (trained in another class or so with the chimp!). It was so stressful because I felt like a "fake" when I worked at a smaller, satellite hospital and had to perform an EKG on a patient who was coding!
I declined a job when I was one of the first who was offered one. The pay was about $1 less than I could make as an office assistant - and as an office assistant, I didn't have to go into lock-up psych, work with HIV, or Hep. But, it was fun learning and the next time I'm in the hospital, I'll ask the person sticking me how long they've been doing their job!
First of all, it would take between three and five minutes to teach Travis the Chimpanzee how to draw bloods, so the issue is moot.
The "licensing" of phlebotomists is another idiocy in an increasingly structured society that requires three years of training in New Hersey before you can become a "licensed" barber.
If the person who drew bloods was acceptable in the mind of a licensed M.D., than than should be "good enough".
As Ernest Gann said in "Fate is the Hunter"......."rulebooks are paper."
I presume you have had a bad experience with a "blown" vein.
Even the most practiced phlebotomist ocassionally has a problem.
If you'd had a problem, I would suggest you cut the physician and his assistant some slack and go on to protest Global warming.