On the back end, rather than Aetna shelling out money to the hospital so that you can get treated for free or cheaply, the government would do so - just like it already does for Medicare and VA.
So it is not $18tn additional burden - it is a similar burden, but not going through the hands of private insurers who take a cut.
Insurance is one of the things that a government can be very good at. It's not an area of great innovation, so the benefit of having it be a free market is relatively limited, and negotiation leverage is very important, so size is very beneficial. The natural tendency of the private insurers has been to consolidate and to look more and more like Medicare in terms of size. Except they also have a profit mandate, whereas Medicare doesn't, so they take a percentage of all spending and suck it out of the system. That portion isn't helpful to the users of the system.