Then, once the cause is understood, the next step would be to figure out an appropriate system that would work, or at least work better, and amend the Constitution to reflect that new system.

The alternative is to just let things continue as they are going, and I don't like that choice. It results in the president becoming more of a ruler and less of an executive. I expect less freedom down that road (and eventually none), and I don't like it.

My take on the questions I posed is this: Yes, it did work, at least adequately, and now it doesn't. What changed? I suspect that it was Roe v Wade. Since then, the right has been trying to get control of the Supreme Court to overturn Roe, and the left has been trying to block them. So both are fighting over control of the nomination process, which means the presidency and the Senate. I suspect that that's the cause of the trench warfare in Congress.

If I'm right, then it is in fact ultimately unfixable. Neither side is going to compromise, ever. You can't even fix it by amending the Constitution, because then there's going to be a battle about undoing the amendment.