It's a good point, b/c single payer does lower costs by essentially dictating the price they'll pay. And presumably if the price they are willing to pay is below what someone is willing to work for, then the outcome is nobody gets that service at all. Which is why price controls often lead to massive shortages and can be miserable failures.
The crux is an issue of trust... there's going to be some "bureaucrat" controlling what gets reimbursed/allowed. Who do you trust to be the one making that call? With insurance companies, particularly in a system where there isn't a lot of real competition (being tied to your place of employment for many of us, and even the exchanges have limited choice) - there is a strong profit incentive to deny costs, and not a lot of fear that consumers will be able to switch to a more generous provider. With a public option, the metric will be something different, though presumably, cost will still be a major driver.
It's been talked about for decades... our current system does not work great. We pay more for worse outcomes than our European counterparts. We pay a LOT more. So clearly there are massive inefficiencies... and we have a dysfunctional system that refuses to correct it.