And you enjoy a lower premium on your health insurance, because your dependent children are on the government program, instead of on your insurance.

Not nearly enough to compensate. The increment for adding children (especially additional children) to an employer health plan just isn't that much compared to the overall cost of health insurance. And you're almost certainly paying enough in additional taxes to more than eliminate even that benefit.

It's just not a recipe for a stable program.

The obstacle to a single-payer system in the U.S. is the disruption and dislocation caused by the transition. There are too many groups (and not just PE "ticks" or health insurance companies) that would be made materially worse off by the move. People hate losses more than they value gains, and it's hard to pass a program that imposes concentrated costs on one group to delivered small benefits to a larger group - both of those things make it very difficult to come up with a politically palatable transition process.