The problem is the position of PM has become presidential. This was started by Margaret Thatcher, before that woman the Cabinet was more collegiate.
Essentially, if we want a presidential-style leader, that person should be directly elected and have an individual mandate.
Alternatively, we could grow up as a nation and treat the Cabinet as a collective, and stop making such a massive deal about the perceived personality of the PM, how he/she dresses, and so on.
Of course, if you want a presidential-style leader the Constitution needs to be changed and there are all sorts of implications. You could, for example, elect a PM who has not got a majority in the House.
In my view, it would be sensible to have party leaders elected by MPs only. In the last analysis, the MPs are best placed to judge a leader from their side and their confidence in the elected person is absolutely key.
What we have now is an utter shambles, neither fish nor fowl.