I decided to take this challenge tonight. I forget what magazine this is from, but I have meaning to try it for a while now. I went in without having played the game in a few months and managed to get through it without losing a single Mario and without hitting the Yellow Switch Palace.
The magazine seems to make a big deal about the difficulty without using the Yellow Switch, but it has very little effect. There is one spot in the first few levels where it could be handy if you are clumsy and fall off some moving blocks, but otherwise it is meaningless.
The most difficult part is finding all of the hidden exits for the first time. I remembered where most of them were from prior play-throughs. There are two that benefit from the Green Switch Palace, but both are doable without the green blocks if you give it a little thought.
I was actually a little disappoint with how easy this was. I guess I have been playing the game on and off for 20 years, but still. I was going to take a photo of my title screen showing only 11 exits, but I accidentally went out a normal exit, so it showed 12 on the screen. I immediately tried another play-through to get only 11, but died because I was rushing and just stopped there since I had already acheived my goal the first time. Besides, there is nothing on the title screen to indicate the game was completed.
Anyway, I suggest that anyone who wants a mild challege, give this a try. If anything, it will remind you of how mind-blowing this game was back in 1991.