The two players in this instance want to find out who is better at the game. When a player makes the first mistake more often than the other, the other is probably the better player. Honestly, even if they did not eliminate the environmental hazards and items the results would still be the same. Having those things included, however, will only get in the way of finding out who is better.
My friends who I played this game with basically all started at about the same skill level and were pretty competitive. So, with items on, we would play and it would be a very close game, but then someone would get a legendary pokeball or a starman or a heart or something that would give an outrageous advantage to one player, allowing them to take victory. To competitive players like us, having a game end in this way is like ending a close game of basketball suddenly with a coin flip; it is an unsatisfying ending. Having all been on the receiving end of having a victory by merit snatched away by chance, we turned off the items. Since the battles after that were decided primarily by who was the better player, our skill continued to improve. We did not spend long hours "training" to become good, it is just a thing that happened.
I realize that people like to play with the items on and it is fun in a more random way like how mario party is fun, I am just hoping to give some insight as to why people like to play with no items, etc.
To answer your question, "Why are people who play only on specific stages (Hyrule Castle) with no items even considered great in general?" It is because the people who are considered great are extremely skilled at the game, and they play with those restrictions to increase fairness to both opponents.