The problem with the "best person for the job" policy is that at the moment it invariably seems to be that the best person for the job is a middle-aged, relatively well-off white British man. At some point you might begin to wonder if this is because he genuinely is the best person for the job or if somewhere along the line there was some prejudicial selection process that excluded an actually better person for the job on some shallow grounds, such as their skin colour, or gender, or sexuality, or whatever other aspect of them that is utterly irrelevant to politics in theory but some people still can't see past.
Or perhaps, now, this is no longer the case mostly but there is still a relative lack of candidates with different cultural/ ethnic backgrounds. This does matter because Parliament shouldn't just be representative of the people, but it should also be seen to be representative in some sense. One can imagine a black youth, who grew up in the inner city somewhere, wondering if he should get engaged in politics, but tuning into BBC Parliament or switching on the news, or some such, and seeing an image of a Parliament which has very little link to his background; and so, he might then think, how can these people possibly be aware of the issues that affect me? He might even be right to some extent, although it's more than just about the skin colours of people in Parliament. But is it really so bizarre to suggest that the people who appreciate the best what it's like to be a young black person, and so would understand whatever unique problems that might mean, would be someone who actually was a young black person?
And then perhaps one way to try to engage people of different backgrounds in politics, to provide a decent first impression that it is, after all, relevant to them, would be to try to ensure that there actually are people of different backgrounds at the highest level of politics.
As long as first impressions depend to some extent on colour, race, religion, sexuality, gender etc., you can make this argument. Of course, in the future, you'd hope that you don't have to any more. But we aren't there yet.