This:

each district should be of similar population size, and every district counts for one.

seems to run counter to this:

The extra steps mean that politicians can’t purely focus on population centers, rural communities would count for the same vote.

As an example, lets say you have a rural area with 1000 people in it, and you decide that each district should contain 1000 people, so that entire area is one gigantic district. Nearby you have a city with 10,000 people, so you split the city into 10 districts. That city still counts 10 times what that one giant rural area does. The only way I can see where you could make the rural area count for more is with extreme gerrymandering where you snake little bits of every rural area in to include a chunk of the city population thereby diluting the strength of the cities vote by smearing part of it over the rural areas.

I see absolutely no reason why we should adopt a system that exists solely for the purpose of making gerrymandering possible, and I see no reason why doing things this way would make any difference over just using the popular vote if you aren’t gerrymandering.