What if there were no districts, and everyone voted for a single candidate out of all the candidates running in their state. Any candidate that could get X signatures gets on a single statewide ballot. Each voter picks one. If there are 18 districts, then the top 18 candidates who get the most votes win.
A natural feature of this is that candidates from the same party or who are highly similar to each other would be competing more directly with each other for votes. It would give an advantage to independents and those who stand out as unique yet still have appeal to a significant portion of the population. It would likely reduce attack ads, because its harder to attack 20 other candidates.
Candidates that appeal to 20% of the States population should get a seat in state with more than 5 districts, yet have no chance under any system with districts, unless that 20% happens to all live in the same place. Under my proposed system such a candidate could easily win.
Alternatively, if we are to have districts they should be drawn at random with the only criteria being to maximize the ratio of the area within them relative to the length of their borders. And they should be randomly redrawn every 4 years, so that any incumbent only gets 2 terms before they must re-win with support from a whole different segment of the state's population.
I live in a state with one single significant metropolitan area. The rest of the state is mostly rural, a lot of it is farm land, and a significant portion is wilderness with some small towns. An overwhelming portion of the state's population now lives in the central urban metropolitan area. However, the state does depend quite significantly on the outlying areas for a number of things and a not insignificant portion of the state's economy is outstate.
Add in that the state is very broadly divided into 3 to 5 distinct areas which each have their own concerns and needs. If the entire state voted with no district representation, the people in the cities would be vastly over-represented and their interests would dominate. Ultimately, this would lead to harm even to the majority as they would ultimately lose or spoil what they rely on from the outstate areas by voting for their short term needs. Easy example: the large metropolitan area has obviously more significant need for road and highway maintenance and construction.
Districts are always based on population size, so any district system will always give much more power to densely populated regions than sparse urban areas. If each person has only 1 vote in a 10 district State, then the urban areas will not win all 10 seats. The rural people can still unify behind particular candidates and give them enough votes to win. If there are 10 seats, they only need enough votes to beat the 11th place candidate who isn't likely to get much support from the Urbanites.
The only solution to the problem you are raising is to give more political power rural people over Urbanites, which is precisely what the GOP is already doing and how they are winning a higher % seats than their actual % of the state's population.
And the problem in NOT actually just gerrymandering by the GOP. The GOP will have an inherent advantage in any system that creates districts based on geography, unless it goes to great lengths to create just the kinds of complicated shaped districts we have now but with a pro-Dem bias.
Part of the problem is that major cities are more extremely "blue" while lots of rural areas are only slightly "red".
If we simply do regional districts, this means that most the "blue" votes in urban areas get wasted electing 1 candidate by 70% when they only needed 51%. But "red" votes in rural areas are able to win more seats because they win by small majorities.
IOW, if 60% of Illinois residents are Dem, but 80% of Dems live in Chicago, then the GOP will win most of the seats in Illinois.
BTW, I am not convinced of the no-district idea, but I don't think other approaches will solve the problem you are talking about or the problem that the Dems are under-represented in Congress.