Last night I beat the game without too much difficulty. My 100 lives helped -- I used up some 50 of them. I found level 8-3 to be quite fun with few precision jumps. I remember I have played level 8-4 before, I think it was on Super Mario Bros DX on the Game Boy Color. But playing on the real hardware is more difficult. I also don't remember that I have had to do the jump trick in 8-2 on the GBC.
I think it is necessary to play this game with a strategy guide for the first time. A lot of necessary things are otherwise unlikely you'll discover.
It turned out I was wrong in my last post. You will play world 9 if you beat the game without using warps, true. World 9 ends with the text "Arigato" (in katakana) so it's to be considered a bonus level more than anything.
However, worlds A-D are accessible if you have at least 8 stars in the title screen, and press A+Start. I did not know that. So I didn't have to beat the game to access world A, because it already had 8 stars. Worlds A-D are considered secret worlds so I don't consider them necessary to beat the game. But I looked at them and they look fund. I don't think they are impossible, having warps, 1ups etc. I may have a try at them later.
I shot bowser dead.