Is there any situation in which a non-governmental 501©(3) would sponsor a 403(b) that includes employer contributions and the plan would not be subject to ERISA?

If you're asking about a nongovernmental, nonchurch 403(b) program, it's subject to ERISA unless it's a pure salary reduction arrangement. You can find more information at DOL Reg. Sec. 2510.3-2(f).

But the main issues are compliance with ERISA reporting requirements, etc. Not all the fiduciary issues.