I know that I am very late in replying to this, and I apologize for that. I have actually been mulling this over since you first posted it, @treyguinn. It has been niggling at me, so I finally decided to answer it.
I am a big Middle Earth fan, so I am going to first address your question from the Tolkien standpoint, as then add a few bits and pieces about Merlin (who is very similar to Tolkien’s wizards, anyway, at least in how magic goes). I’m a fan to the point that I would totally write a grimoire about the Valar if I wouldn’t get sued by the Tolkien estate. Now, there is a lot more to Middle Earth than is shown in the movies, so I’m not sure if you’re familiar with the books and mythology of the landscape or not. I’m going by the books, because they’re killer.
To answer your question, YES, you an totally become a wizard like mythological ones. The reason there are stories about them is because they existed or they are based on people who actually existed. As you noted, mythological wizards did not make sparks come out of the staffs or turn people into toads (goodness knows I would love to turn someone into a toad). They were practitioners of magic in the real sense of the word. They were followers of a Left Hand Path, they sought greater wisdom, knowledge, and power than those around them and they succeeded in getting it. And they worked damned hard to get it. And you will have to also, if you wish to be like them.
Tolkien actually had a fantastic grasp on what magic was and what it was not. The Istari (Tolkien’s wizards) are actually a great thing to aspire to, especially in a forum called “Become A Living God”. 
The Istari, of which there were five, were ainur, which were lesser gods. We could compare them to the multitudes of angels or demons that we work with, greater than a genius loci but not as great as something that was worshiped as a god in its own right. They took on bodies and came to Middle Earth to help steer it in the ‘right’ direction (the definition of right being debatable and not the subject matter here). They used their personal, divine power to create changes in reality. Just like ‘real’ magic they couldn’t do anything that was ultimately against the laws of physics, both known and unknown. If you read The Lord of the Rings, and pay attention to what Gandalf actually does and the results of what he does, you will see this clearly. And is our goal not to become a god/ess ourselves? To move our own worlds in a way that we control everything around us? That is exactly what the Istari were able to do. It was only when they doubted themselves that they were caught in the trap of their own making.
Merlin was half fey, one of his parents was a faery. So, he was something very similar to the multitudes of angels and demons that we work with as sorcerers/esses. In fact, several on this forum work with the fey directly. He manipulated the world around him with his magic to change it to serve him in the best way possible for him. There are stories of him changing into animals which some sorcerers/esses might think is not possible, but there are a slew of them who do feel that it is possible, that do it themselves, and would tell you that Merlin, if he existed, did it too.
My point in all of this is yes, you can be just like them. Successful medieval magicians, just like these mythological personages, understood how the universe worked, even if it was within the scope of their times and mythology. They understood it to such a fine degree that they were able to manipulated it. Most of our magical practices come from these people! Have you read the ancient grimoires? It was these medieval magicians who wrote them. They knew what they were talking about–ask anyone who has worked through the Goetia.
I won’t go into to the how, as you didn’t ask that, and this post it probably already long enough. Hopefully this helped.