That's kind of already the case, though? When the MP is of a party that's in government, they probably already get more letters from constituents, because there's an expectation that they can do more because they have more power (often not actually the case, though). But people have all kinds of reasons for contacting an MP, and all kinds of criteria for selecting who they'd rather deal with: whoever is geographically closest if they need an in-person appointment, whoever is in the right party, whoever has voiced an opinion the person agrees with, whoever has spoken up on the issue the person needs help with... It would all balance out more than you'd expect.

The problem with not having it district-wide is that say one party got an entitlement to one MP. That would mean that over the entire region, they got approximately 15% of the vote. Over a constituency with 6-8 MPs, it's fine that 15% of the population get represented by 15% of the MPs. But if that single MP is assigned to a specific 1/6th of the constituency, that constituency is then 100% represented by a party that only 15% voted for, which is actually less democratic than the current system at a local level, even if nationally it's more representative.