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Even better: have representatives who represent people.

We have proper information systems, we can do this. A representative gets qualified by having the appropriate minimum number of people sign up as their constituents. They then represent exactly those people.

For all the examples of doing geographic districts I sometimes wonder why is that specific special interest group that gets privileged with a district separation instead of any other way to split people. Prussia split districts by wealth too but that was a way to give more power to the rich.

How would you section off the list? If you do it uniformly randomly, then every constituency looks identical and you'll get identical results in each constituency. This creates a very non-representative result. Notably, large racial minorities would have their voting power diluted.