Your town assembly can take care of the concerns of your town.
Your regional assembly can take care of the concerns of your region.
Your National Assembly can take care of the concerns of your nation.
We can get rid of gerrymandering and still have different levels of government, each devoted to a particular level of localism.
Indeed, having one level of localism interfere in the affairs of another level of localism seems like a violation of encapsulation, in the sense that a software developer would understand the concept.
Another thought I’ve had for a long time: why not just lift the limit on Congress? My understanding is that Congress set a law to cap it (as opposed to the constitution specifying it) Which made sense 100 or even 50 years ago when members of Congress needed to be physically present to vote in DC.
But today in a world of the internet, video conferencing, etc. it seems like that limitation is arbitrary. Much smaller districts would exists. Making gerrymandering harder. Making a the house look a lot different than the senate. It would make the House more like … a mayor who writes and votes on laws in DC instead of locally.
Congress could easily double the size of the House, and should. That and a law requiring that districts have maximum compactness/population density would I think solve many of our problems with running a democracy.
There's a certain appeal to that too however. As long as the advantage/disadvantage isn't overwhelming anyway. Perhaps it would help to encourage more diverse communities or give a disadvantage to political groups that isolate themselves while giving an advantage to groups who demonstrate they're both willing and capable of living and working in diverse environments. It might even help to reduce the amount of political and cultural divide.
Note that in order for this approach to be coherent at all, you need to abandon the idea that it's possible for a voter to vote for a candidate for office. The only kind of vote this system acknowledges is a vote for a political party.
(As a side note, Wikipedia's toy example of "STV in a party system" considers a case where three parties field candidates for 5 seats and receive 48%, 45%, and 7% of the vote, resulting in those parties respectively winning 2, 2, and 1 seats. This is about as bad in terms of proportionality as any other system [Party A has 686% of Party C's votes and 200% of the representation; Party B has merely 643% and 200% of the representation], which isn't a surprise; 5 seats don't allow for proportionality unless you're lucky enough to see a 63-37 split in the electorate. But if party-based proportionality is a concern, the correct answer is fairly obvious - the three parties should receive 2, 2, and 0 seats, with one seat going unfilled. It is interesting to me that no one ever seems to contemplate this as a possibility.)
I don't think that the approval voting scenario you suggest is good though. If I want to elect five candidates, I want 5 candidates that represent the diversity of the population. But it seems to me that your suggested method will elect 5 candidates of the largest voting bloc, not at all what we want.
The example is very explicit that votes are cast for parties and all of the benefit of receiving a vote goes to the party for which it is cast. As such, the model is only coherent if there is no such thing as voting for a candidate.
But really, what exactly the original example intended to convey is a pointless pedantic argument I'm not getting into.
The crucial point is that you absolutely can have multi-member proportional districts where you vote for specific candidates. So to someone who advocated for a such multi-member districts, the contention that you have to give up voting for individual candidates simply isn't true.
Just increasing the number of representatives would have some impact on gerrymandering.
Proportional representation solves this, but if you still value the notion that local communities should be able to have their collective power represented, then you need a solution that combines the two.
As mentioned elsewhere in the comments, multi-member districts with proportional representation are a good way to do this without needing to change the Constitution. The gist is, split states into a smaller number of larger districts, each of which gets multiple representatives. The representatives are allocated proportionally to the vote in each large district. So if the district has 5 representatives, party X gets 63% of the vote, and party Y gets 37% of the vote, then party X gets 3 members and party Y gets 2 members.
Political scientist Lee Drutman is actively promoting this approach, and recently founded Fix Our House (https://www.fixourhouse.org).
If I could rewrite the Constitution, I'd get rid of the rounding error by weighting the votes of individual members of Congress. E.g., if there were only two candidates in a district, candidate X got 60% of the vote, and candidate Y got 40% of the vote, send them both to Congress, with X getting 0.6 of a vote and Y getting 0.4 of a vote.
Of course all of this is better when combined with instant runoffs (aka ranked choice).