It seems to me that a system with zero efficiency gap would be equivalent to one big election with proportional representation. If that's what people really want, why talk about voting districts at all?
Something that you can do is something like we have here in Sweden - we have a district setup with some places for each district that is proportionally distributed.
But that create inefficiencies as that heavily favour some parties. So we have a chunk of undistricted delegates that are given out to even out the overall proportionality. This get you pretty good results as long as the amount of parties stay within reasonable boundaries[].
[] Which they of course haven't - we currently have 8 parties in the Riksdag.
Because certain issues are geographical in nature, especially at the local level.
Should a new highway replace the housing development over there? Well, people who live close to that housing development might care one way or the other (Noise complaints, Ecological issues, Traffic Issues). People who live very far away from that center probably don't care at all. And the people who live in the housing development (who will be forced to move out), especially renters who probably won't be properly compensated, will care the most.
The placement of schools, the budgets of police, the design of zoning regulations (in particular "Enterprise Zones" of lower taxes to encourage business development in some areas)... these all are innately local issues.
If I had to guess, I would say that there is at least some benefit to having politics be local. That is, it may be better to ask people to choose between a small number of "local" candidates, rather than a large cohort of state-wide candidates.
You're either thinking about this too broadly, or as a system filled with static entities. Within voting districts, politics is local - individuals can sway voters, voters can decide to show up/skip voting, etc.
Also, the concept of a 'blue/red' district can be very blurry - there are many examples of 'blue' districts (districts with democrat congress reps, and a history of voting blue) that voted red in the last presidential election, and vice-versa for previous elections.
Historically, Americans prefer to vote for people, not parties.
But that create inefficiencies as that heavily favour some parties. So we have a chunk of undistricted delegates that are given out to even out the overall proportionality. This get you pretty good results as long as the amount of parties stay within reasonable boundaries[].
[] Which they of course haven't - we currently have 8 parties in the Riksdag.