So our system is perfect, no improvements can be made other than throwing more money at it. There is nothing wrong with a two tier system as long as it doesn’t take resources from a public system.
I think the problem with a 2-tier system is that there is no proper public to govern the government on how they provide services. Right now, it's a top-down system and any calls for reform are met with comparisons to the terrible US system. If we were to radically change our system as you are describing, we would need to form some public sphere that would understand such concepts as service levels, tradeoffs etc. We don't have that now.
If we did, I would say that not only would a 2-tier system be possible but it could be used to fund improvements across the board. If Joe Richguy wants to spend money to go to the front of the line, why should I care as long as my line gets shorter ? I understand the concept is abhorrent but if we framed it differently we could accept service differences and fund service improvements.
Part of socialism is actually making the rich pay, not resenting their existence. Two cents...