I am in favor of a mixed system that combines single member districts with at-large seats to fill out proportionality. I think this combines the ideal of local districts where individuals are accountable to their constituents, with the political reality that most politicians are an empty suit for their party. In one regard, this will allow for more party loyalists as about half the legislature will be elected by a slate selected by the party than as individuals chosen by their constituents. On the other hand, it is probably the best means of allowing multiple political parties, which hopefully sees the Republicans and Democrats fracture, decentralizing political power.

This helps to alleviate gerrymandering. While individual districts can still be gerrymandered, and gerrymandering can still dilute political power of communities, the at-large seats minimize the effect. Gerrymandering itself can never really be eliminated. In a single-member district, where the population of each district must be the same, there has to be some choice made on how to draw boundaries, and those boundaries will almost always arbitrarily split up geographic, cultural, ethic or racial communities. In an at-large system, while you can't gerrymander in the same manner, by eliminating districts, you are reducing local representation and accountability which can have a similar effect to gerrymandering.

I don't think such a system necessarily gets rid of any sort of extremism. I think you'd likely still have some form of primary elections where constituents vote for the person they think best represents them and their party. However, by breaking up the prisoner's dilemma of first-past-the-post single-member districts, there is no longer an incentive to maintain a big tent party where half the party is holding its nose while voting simply because the other guy is worse.

By making multiple parties possible, the strategic voting incentive is almost entirely reduced.

For single-member districts and unitary offices like governor and president you can use ranked-choice such that the winner has to have the broadest coalition. This allows the voter to vote their conscience first and foremost, as doing so has no cost. In a first-past-the-post system where the winner only needs the plurality of votes cast, voting one's conscience can effectively be throwing away a vote. In a ranked-choice system in which the winner needs to get a majority, there is no spoiler effect. In the former system, candidate A gets 40% of the vote while candidates B and C get 35% and 25% respectively. Candidate A wins the election. In the latter system, the same split does not decide the election. Instead candidate C gets dropped off the ballot, and each person that voted candidate C as their primary choice will have their secondary choice counted instead. Lets say candidate B and C had similar enough views that 16% of C's voters chose B as their secondary candidate. B wins the election. No voter was injured by voting their conscience first and foremost.

I will add that in addition to all of this, the goddamn House needs to be expanded. This can happen with simple legislation. I also believe the Senate should be doubled, but that requires a constitutional amendment.