Reagan probably wouldn't have been "Reagan" under a parliamentary system. If the US had the Australian constitutional system, then, for example, a Democratic parliamentary majority probably would have enacted universal health insurance in the 1960s or 70s, and we'd still have it today.
What I'm getting at is that since the Democrats are the party that's more interested in activist government on economic issues, their agenda suffers more in a system in which there are many veto points. The American constitutional order is "conservative" in the sense that it tends to conserve the status quo. Big social programs are hard to pass. So if the USA had a parliamentary system, then I imagine that the political spectrum would be shifted a bit to the left of where it is now, at least on economic issues.
Of course, there are all sorts of other confounding issues, like the fact that individual members of Congress act as free agents in a way that doesn't happen in most parliamentary systems, where things are run in a much more top down manner. Legislative power is incredibly diffuse in the US.
I think you are sensing a fundamental outcome of the American Revolution. A major point of contention was the top-down control and ease by which regulations could be imposed on the colonies by British government. Their constitutional solution was a diffusion of power with significant checks on power exerted from any one branch.