The issue with that system is while there is checks and balances available, One Branch in particular holds the majority of the authority in that balance system, and that being the legislative branch, and when that Branch doesn't work together you might as well throw that checks and balance system out the window.

We are seeing this issue with the current Supreme Court where it's very clear that there is visible corruption in the seats, but the executive branch can't do anything about it because the legislative branch can't get along long enough to be able to do it. This is exactly why we are warned against having a two-party system in the first place

Our system while on paper seems nice falls apart instantly the second any of the three branches decide they no longer want to do their job, or can't agree with an outcome,