“The poorest of society (while the pool of altruistic donors is eliminated)”
Correct, and the poorest will be much wealthier as a consequence, better able to feed themselves, get good accommodation and give their children a good start in life. After all, it will be their choice to do it, so they won’t if they don’t think it is worth it.
As for altruistic donators, why do you say that? Does paid work stop people from volunteering? Do pharmaceutical companies stop donation to medical research? No, of course not!
“Furthermore, the organ donating process is currently free at all stages of the process, including the medical professionals who deal with the donation process but do not get compensation. So what's going to happen with them? Are we going to have to compensate them too?”
If they wish to demand pay, they can, if not, they can do it for free. At every stage it should be the person’s choice that matters.
“Is life a right? Saying people have a right to life but health is a privilege is like saying people have a 'right' to self-defense but owning a weapon is a privilege. Now, usually the libertarians re: health are adamantly pro gun "rights", so I'm curious as to how this argument works.”
The right to life is not the right to have your life supported, but rather the right to not have it taken away from you. You have the right not to be actively killed, but not the right to be saved from death- that is a privilege that people can grant you.
“For profit organ donation would bring about a nightmare situation. The medical society already has far to much financial pressure on it already...”
This is emphatically untrue. The insurance model of healthcare that the US government mandates is causing healthcare providers to not be under enough financial pressure. That’s why heathcare costs are not dropping as you would expect, except in areas not covered by insurance, e.g. cosmetic surgery.