Decoupling health insurance from your job and going single payer are separate things. You can have one without the other. You can have a very healthy system with private insurers (like in Switzerland) and you can have completely dysfunctional single payer system (like in Poland and I guess many other countries).

Main problems are elsewhere: doctors gatekeeping the market, lack of transparency when it comes to pricing, regulation making it impossible to compete for smaller players etc.