I've never heard of such an institution either. But I can imagine a reason why someone would start one. By analogy, some musicians have started free music schools via charitable foundations. But these are typically for children in poor neighborhoods (and other select groups in need). The musicians are passionate about music and feel strongly that everyone should have the opportunity to learn it. Here's one example:
https://www.sarahschoolofmusic.com/ .
So there would need to be physicists that have a passion for physics and feel strongly that everyone should have the opportunity to learn it. Not sure anyone would launch an independent school though. Physicists in this category would most likely endow scholarships or fellowships for study at existing universities (to teach physics properly, you need more than physics classes; you need labs, as well as classes in other subjects such as math and computers).