If memory serves, Dan Jarvis also did it, being both MP and mayor of the South Yorkshire city region or whatever it was called at the time.
It is fairly innately political. No Prime Minister has ever polled as low as Starmer and come back from it, or so is being said in the press. Burnham might be a smart electoral move, but he's not a plaything of the Labour right, so they kept him out.
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.
It's actually the role of Police and Crime Commissioner that prevents them from being an MP simultaneously. In Greater Manchester (and London) the PCC role is combined with that of Mayor, but it isn't in most other city regions.
There's not much actual difference in the mayoral aspect of the roles - Jarvis was the Mayor of the South Yorkshire Combined Authority, not simply the mayor of Sheffield City Council.
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.