<PERSON> probably did not approach the question as you did, as from the logic of the world history of race creation, the character of a wizard was probably very close to him, in the form that it is told, as a central magic human shaped figure, that anyone could aspire to be. He probably had enough to think about regards elf scripts, casts, the planning of his story, to have devoted much time to the question of what race the wizard should be, the wizard being a good centre for the narrative, and being the most most powerful character in the book.
It is understood that the wizard is not strictly human, and the most information you get regarding this, is <PERSON>'s words regarding the origin of the wizards:
<PERSON> was one of the five Istari sent to Middle-earth by the Valar in the Third Age. In Valinor he was known as <PERSON>. <PERSON> was instrumental in bringing about the demise of <PERSON> in T.A. 3019, chiefly by encouraging others and dispensing his wisdom at pivotal times. <PERSON> was originally robed in grey, and second to <PERSON> in the Order of wizards. 381
After his fall in Moria, <PERSON> returned to Middle-earth as head of the Order, robed in white. <PERSON> was noteworthy for his keen interest in Hobbits.
Men:
The race of Men is the second race of beings created by the Supreme God, Ilúvatar. Because they awoke at the start of the First Age of the Sun, while the Elves awoke three Ages before them, they are called the Secondborn (Quenya: Atani, Sindarin: Edain) by the Elves. Men awoke in a land located in the far east of Middle-earth called Hildórien. Men were created after elves, but may have been created prior to the wizards.
In the Hobbit, the first description of <PERSON> appeared:
...an old man with a staff. He had a tall pointed blue hat, a long grey cloak, a silver scarf over which a white beard hung down below his waist, and immense black boots.
Note that the Hobbit wasn't written entirely planning forwards to the later books. It is much simpler regards, world, history, etc, the later books are a more advanced vision. 381