I think you need a Republican President who forces McConnell to compromise with the Democrats. I'm not a McConnell fan. Perhaps the same could be said of a Democratic President and Schumer.
Presidents have very little direct power over Senators.
Over House members either, but especially Senators.
I don't think the two party system is the sole factor for polarization. I think the construct of the legislature and the electoral system are also a factor. Our Federal political system was designed to make change a very slow process. We often have Presidents who do not control the legislative agenda. Instead of forcing compromise, we have had for many years obstruction from both sides. The favored strategy for the party out of power has been to wait an election cycle or two and be returned to power.
This is all true.
The US is particularly laden with veto points that can stop things and it makes pure obstructionism a much more available tactic than it would be in many other systems.
Like two parties though, this isn't in and of itself the cause of polarization, but it does help incentivize it.
It's hard to say if this polarization is the natural default for the U.S. system. It could just be the current state as long as the two parties have a rough balance of power.
Historically, it seems to have swung back and forth a few times.
There was even an time called "The Era of Good Feelings".
It looks more like a case where because the system allows you to be quite effective as a polarized opponent (provided you are fine with blocking things more than accomplishing things) then in cases where polarization is high or there is a particularly hot issue, the structure allows for a feedback loop where polarization is effective and that brings about more polarization.