There are some nifty things wealthy people can do with a 501(c)(3) with regard to tax advantages, but opening a clinic just to get PSLF status isn't worth it. As pointed out, you can't distribute profits to yourself. I suppose you could hire a spouse or adult child (wage + retirement benefits) with the profits, but then you have W-2 taxation. If you go through the work of starting a clinic, might as well make it a PP and benefit from salary + profits/dividend taxation.
PSLF itself doesn't even make sense (unless maybe you are in academia) because many employers offer loan repayment or you can pay off loans relatively soon by joining a PP.