Nope, I don't need saves to be able to complete the four levels at least on non-hard difficulty. I feel I might've passed them on hard as well, but not completely sure. Instead I'm pretty sure I wouldn't manage the 4 levels with Jazz without saving. Especially 'Rocky Ride to Nowhere' is much harder because Jazz kinda "slows me down." Hard to explain it better.

The main problem with 'Astroid City' are the bomb crates that have the coins needed for the warp. The bottom level where you have to shoot down the crates and open them in under half a second before they fall down to the pit. The bomb shockwave usually pushes me right into the pit even with Jazz'es uppercut. If I was able to do this, then I could as well finish the level. I don't know though. Maybe I should try more. It was about 5 years ago when I last time played that episode in a serious manner.