The American University, mine and Mrs Vache Folle's alma mater, is doing very well in competition with the University of the District of Columbia and the University of Maryland, two nearby public options. TAU is a private entity that competes head to head with public entities for students and faculty, and I haven't heard anyone argue that public colleges will drive private colleges out of business.
FedEx and UPS don't seem to be much put out by the existence of the United States Postal Service.
Public and private hospitals coexist in many communities as do public and private golf courses and other recreational facilities.
Public housing exists right alongside private apartments.
The MTA runs trains and buses in competition with private cabs and limos.
Why couldn't a public health insurance option coexist with private health insurance? Is it because we don't want people to have too many choices?