Districts by their very nature represent voters.

I feel like you are misunderstanding representative government. There is value in districts, provided they are drawn apolitically. Without it, people living in sparsely populated areas would effectively have their unique needs unmet.

It's really important to understand why this is not the case. Districted voting essentially introduces first-past-the-post voting at more levels. Each level of FPTP creates a larger disparity between what voters want and who gets elected. This is in part due to gerrmandering, but that's not a required thing.

Every time you decide a district election through FPTP, you essentially create a rounding error, a disparity between the election results and what voters actually voted for. This FPTP system then reinforces the two-party system that the US and UK have a very hard time escaping. And as you may be able to guess, having a mere two major parties to choose from is fucking terrible for getting niche voters represented. It's why the US and UK see comparatively little regional focus and increased disillusionment with national politics in these areas.

Abolishing districts actually helps local representation(!). Because under proportional representation, if someone campaigns on serving the needs of a small group of voters, said group can vote for them and they will be elected. It lets anyone basically define their own "district" of voters, without political manipulation. If they fail to attract a sizeable enough share of votes, then this electoral niche is simply too small to be represented at the national level, and this group should perhaps petition local government instead.

We see this effect quite clearly in countries like the Netherlands, where there are quite a few national parties to choose from, and several focus on a specific group of voters (eg the BBB which focuses on farmers, or the FNP which focuses on people living in the region of Friesland.