Also these districts could use ranked voting (single transferable vote) so could avoid most of the negative aspects of the current system.

One way to start with this would be to create "fantasy legislatures" based on this kind of districting / voting. Initially these would be just a hobby but if the process worked it would draw people in and become a way for under-represented groups to get a voice. Maybe it could become a stepping stone to electoral politics and possible even influence reforms to the system.

I'd be happy to work on this if others want to as well. Or possibly it is already underway, if so provide a pointer.