Not quite as clear cut as you imagine. No one can even be considered for appointment in any chief officer rank without having completed various training etc at the college of policing.
Who is responsible for appointing all the executives of the college, that would be the Home Secretary, who is the boss of that organisation. As such she or he has a very real influence in shaping policing at all levels.
So there is effectively an approved list of those who can apply for such posts, controlled by an organisation run by the Home Secretary. If your name isn't on that list you cannot even apply for such a post.
The Home Secretary also directs the policing priorities of police forces, and can impose meaningful sanctions on any Chief Constable or force that doesn't comply with his or her wishes.
I think it would be hard to argue that dictating what areas of crime the police have to prioritise does not have a very real impact on operational policing.
So if we are talking about day to day issues, what officer patrols what area etc, then you are right.
But the bigger and more important the issues become the greater the influence of the Home Secretary.
To believe otherwise is very naive.