The downside of the non-profit H1-B visa is that you are subject to the “lottery” system. You may or may not be approved. However, you are not subject to the “cap” and can change employers at any time if you are approved for the visa. In the end, if you are approved for the non-profit H1-b you’ll be “cap-exempt” for at least 6 years until your visa expires.
You are correct that if you decide to go for a “for profit” position, that employer will have to file a visa on your behalf.
If you would like more assistance, please contact an immigration attorney. can connect you with a highly skilled attorney that can walk you through the potential outcomes of your situation. . Be sure to check out our webpage to learn more about our services and to connect with us for a consultation.