The rules are inconsistent. You can be Mayor of Sheffield and an MP at the same time but you can’t be Mayor of Greater Manchester and an MP.

That's not inconsistency in the rules, that's inconsistency in what being the mayor means. In Sheffield it means you show up wearing funny clothes every so often, in Greater Manchester it means you have a full-time job, a large budget, and actual responsibilities.

For our American brethren, it's like the difference between being the Mayor of NYC vs the Macy’s Thanksgiving Day Parade King.