Gerrymandering will exist no matter what you do, including nonpartisan map committees, because what counts as gerrymandering is an opinion. We gotta just leap-frog that problem and move to multi-member districts.
Except it doesn't, because you'll end up boxing out voting populations that are significant, but spread evenly and thinly across your whole legislative area. If there's a voting block that is at 20% everywhere, they will never elect their preferred candidate, because they'll never have a majority in any district. Gerrymandering will always be a problem with single-winner districts, because the definition of fair districts has multiple inputs, and there's no consensus on how much priority to give to each.
If that 20% is evenly distributed everywhere, then they don't need their own local candidate. That's like having the men's candidate or the left-handed candidate.
They won't have any candidate. Regardless, the same problem applies. If these people are spread out unevenly, there will still be voters in districts without representation. Their rep won't give a shit about them because they vote for a different team, and the rep on their team but in a different district will care mildly at best because they can't actually vote for that rep.
As long as you had single-member districts, there will be a significant fraction of the voting population who have no one they can lobby who will listen. If I'm a Republican in a Democrat district, I don't have representation.
I literally said the solution in my first comment? Multi-member districts. Each district has, say, five representatives and they're elected using some sort of proportional representation. Sequential Proportional Approval Voting is probably the best for the US. You can read up on the specifics of that method if you want, but in general any proportional method tries to take into account the fact that once a candidate gets into office, the people who voted for that candidate now have representation and some amount of satisfaction, so other people's opinions should be more heavily weighted when awarding the next seat. It's easiest to explain with party-based methods, but essentially, if the vote totals are 40% Team A, 40% Team B, and 20% Team C, then the winners should approximate that vote breakdown. In this case, 2:2:1. What it means is that minority populations are much more likely to have someone in office who faithfully represents them, but majority populations are still going to have the appropriate fraction of the seats in power.