You wouldn't go to a Whitehall department, because Whitehall is even more hermetically sealed against the public than Westminster is. Tell the Department of Whatever that you don't like what they're doing and they will laugh in your face.

The idea is that you go to your MP because your MP is allowed to deal with the relevant authorities, while you most certainly aren't.

Local councillors are mid-way between the two. They get some local power over local issues, but they still have to defer to Whitehall and Westminster on most issues.

So it's actually easier to talk to an MP because s/he will be one point of contact and can find the correct pressure point. If you try to find that point yourself you're more likely to end up being bounced from one place to another, with no one taking an responsibility - never mind bothering to answer your letters and emails.

Metaphorically, people in the UK are subjects, not citizens, and the political environment still reflects that.