Yes. We've been doing that for nearly a hundred years. And the reason is simply that the hospitals don't want to (or can't) pay a market wage for these workers because of the high overhead caused by doctors' salaries, and because of the power that the physicians weild.
Lets rephrase this idea from yesteryear's to present. The hospitals can't afford to pay nurses more b/c of all the insurance premiums it pays for possible malpractice claims. While drs still make a decent living it's NOT what it used to be and most would be better going into something else(ie pediatricians and internists). The insurance companies have lowered drs. wages/reimbursements while at the same time continuing to raise premiums on individual policy holders. The individuals blame drs. for rising cost and the insurance companies laugh all the way to the bank. It's a brilliant scam.
Where are you getting your info? As a general rule, only government hospitals pay doctors salaries. Private hospitals don't. Doctors in private practice bill patients or the patient's medical insurance carrier.