It would only be a mistake if there was actually a TOS episode that explicitly established that the character of Chekov didn't join the crew until the second season. ("Welcome aboard, ensign."). Since there is no such episode we can safely assume that Chekov was among the roughly 430 random crew members aboard the ship during "Space Seed."
Chekov was a lower decker who got promoted to navigation after his first year ('cause he's a whiz kid). His best friend was Boimler's grandfather. During Space Seed, Chekov was assigned to bring Khan his dinner. There. Done.
Edit to add: this also explains why Boilmer is so intimately familiar with obscure trivia about Kirk and Spock's escapades. He listened intently at family story time.
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