Would Twain’s story be as effective if there was no use of dialect?

It's a very good question, and I would certainly grant that writing in dialect can serve as a barrier to entry. That being said, I think this technique is used very effectively within this story, and that the story itself would not work quite as well without it.
First of all, writing in dialect adds an element of authenticity to the story. "The Celebrated Jumping Frog of Calveras County" is ultimately a depiction of everyday life, and trying to encapsulate everyday patterns of speech is a highly effective tool in the realist's repertoire.
Additionally, Twain's story here is shaped by a very powerful contrast between the erudite narrator, who speaks in very precise and very proper English, and the much more rambling narrative told by Simon Wheeler. Through Twain's use of language, one might get the impression that the two men might as well be from very different worlds. If we remove that use of dialect, this contrast goes away.

To some extent the answer to this question is up to individual reader opinion. There are times when I feel that the use of dialect is detrimental to the story. This occurs when the dialect is so difficult to read and understand that it prevents a reader from being able to engage with character, plot, theme, or some other aspect of the story. If the dialect prevents a reader from understanding the story, then I do think that a story can actually be more effective without dialect. With this particular Twain story, I believe that the dialect enhances the story, and I feel that the story would lose much of its value and character if the dialect were to be removed. The story contains a narrative about men betting on a jumping frog. It's a story about simple country folk, and Twain's dialect helps to sell those characters to readers. Those characters wouldn't be as developed or fun to read if they weren't speaking in their regional dialect.

The feller took the box again, and took another long, particular look, and give it back to Smiley, and says, very deliberate, "Well, I don't see no p'ints about that frog that's any better'n any other frog."