It seems you've answered your own question - they'd be significantly lower with single-payer.
But to expand, the US has, by far, the highest cost per capita for healthcare at just a bit over 12,000 dollars a year when compared to similar countries. A large part of this is due to our inefficient system, in which administrative costs are astronomically high:
"administrative expenses account for approximately 15% to 25% of total national health care expenditures, an amount that represents an estimated $600 billion to $1 trillion per year of the total national health expenditures of $3.8 trillion in 2019."
There's multiple fundamental problems with the healthcare system that causes this. Again, these problems are fundamental, meaning they cannot be resolved within the constrains of a private sector insurance industry:
1. Insurance billing complexity is extremely high. There are thousands of insurers nationally, and providers are forced to navigate through that to provide adequate care. As a result, many private practices have gone out of business, because doctors spend more time managing insurance than patient care.
2. Insurance has an incentive to deny as many claims as possible. There's simply no way around this fact - the only way to remove this incentive is to remove insurance.
3. Insurance is not transparent to consumers and consumers typically have little to no choice, killing competition and allowing price gauging.
4. Providers are incentivized to charge as much as the most premium insurer in their area allows. For example, if insurer X covers 650 dollars for a treatment but insurer Y only covers 150, then the provider must charge 650. Otherwise, they leave money on the table for patients with provider X. This means that competition in the insurance space doesn't lower prices - it can only raise them. Yes, as boutique plans for employees such as CEOs gets better, your care gets more expensive.
Obviously they'd be lower with single payer, especially on the provider side, but outside of that the system runs pretty efficiently for what it is.
On the provider side, they're going to have to code up a claim.
But once they're submitted, the majority of claims automatically flow through the rest of the system with low human touch.
That's literally the only way insurance companies can function under ACA admin cost caps.
When that generates a denial... then obviously there's a lot more touch during the back and forth.