//// I'd leave it on, unless you've any reason to suppose the roof underneath it is not in good condition. ////
.... that's exactly what I'm thinking.
From the inside of the shed, the wooden slats seem fine, apart from 2 next to the door, where the felt has come away and has exposed the slats, so they've rotted a bit. I can also see bits of the slats through bits of the tatty felt and the bits I can see look okay.
Just in case it makes any difference, the shed is 8' x 6' and its roof is like an inverted V - similar to the one in this picture:
https://www.sheds.co.uk/adley-5-x-7-windowless-overlap-apex-shed.html?gclid=CjwKCAjw-dXaBRAEEiwAbwCi5vZ5AVLRXjVGQL2U-nOWG7-L-VuT1BGztlHAggjFI60dUDehYoYDrRoCyfkQAvD_BwE