The 501C 3 model just doesn't work in the long run, as it takes to much money to run a successful operation and when you get anyone from the government archaeologist pool you will never get anything done for lack of experience and knowledge of what to do in the water. Just saying
There are a number of long-running archaeological non-profits around the country. Many of them are aligned with museums, and some are more forward-facing than others. I know of at least one non-profit that is active in the cultural resource management sector, and their work has resulted in many books, articles, presentations, etc., though I suspect there are more around the country that I just don't know about.
Every operation and project requires money, successful or not (and depending on what "success" means organizationally and professionally). One can raise money through investors or debt, both of which assume significant risk and require a profit motive. Private and government grants are an effective funding mechanism for non-profits, and since archaeological non-profits are often associated with museums there is sometimes a built-in donor base willing to contribute. There is actually less inherent risk in the non-profit side of things because no profit motive is involved or implied. Exceptional archaeology funded by grants is happening every day, and all you have to do is open a National Geographic or Archaeology Magazine to see it. Government archaeologists are usually tasked with a pretty specific set of annual goals established under law, so their roles can vary from state to state and agency to agency. There are plenty of federal archaeologists that are mostly contract managers and do QC on reports completed by consultants. Others are literally in the trenches actively working.
Then there's the cultural resource management world, where clients hire archaeologists to meet regulatory requirements. There is actually AMAZING work done all the time conducted by contract archaeologists, but it doesn't make the news as much. I think there is a lot of potential for integrating non-profits with the for-profit CRM businesses, but it would likely require substantial front-loading of staff, overhead, etcetera and there probably aren't many companies out there with the margins to make that happen.
Point being, there are great archaeologists doing outstanding work, today, in public, private, and non-profit sectors. I wouldn't say one sector is inherently better than any other (I've worked in all of them). They're driven by different goals, and sometimes do things differently, but to paint with a broad brush and say that someone is inexperienced or ineffective because they work at XYZ instead of ABC isn't a statement that holds water in my opinion.
If anyone wants to talk about this stuff in more detail, maybe a new thread is in order. We've strayed pretty off-topic and I don't want to rub the moderators the wrong way!