Being cynical i would say it's because Burnham could potentially challenge Starmer. Less cynically Labour has a big enough majority they can afford to lose this by election. The headache of replacing the mayor of Manchester is not worth it.

Why can't he just do both jobs? Boris did it iirc.

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.