Star Trek: The Original Series had a formula that worked for it at the time. The main focus of the show was on Captain James T. Kirk [William Shatner] leading the Enterprise into space battles with Mr. Spock [Leonard Nimoy] being the voice of reason and the font of scientific knowledge. Dr. Leonard "Bones" McCoy [DeForest Kelley] was usually part of the three, playing wonderfully off Spock's irritating logic. But in the three years the series was on the air, we never really got to know the characters.
Their pasts came out in bits and pieces, with the exception of Bones, who we never really got to know at all. DeForest Kelley said essentially the same thing in an interview with Dan Madsen in the July/August 1995 edition of Star Trek Communicator, the magazine of Star Trek: The Official Fan Club. "I never felt that he [Bones] was utilized to his fullest," said Kelley. And there had been talk about changing things up, according to Kelley.
"There was some kind of talk in the second or third year of the series that they might start to alternate the focus of the stories. One week would be Kirk, then Spock, and then McCoy," Kelley said. That would have been a great way to get to know the characters more, and the other Star Trek series did start doing that as time went on. Unfortunately, that never came to fruition with The Original Series, and we had to be content with the little bits of information we were given throughout the course of the three years and eventually into the movies.
Today's Star Trek series do alternate stories and give us more background on the characters, but with fewer episodes per season and more characters to focus on, even that doesn't work for every season. The Original Series mainly focused on three characters so it would have been easier to fill in the backstories had the writers/producers chosen to go the route Kelley described. Out of the three, the only one we really got to know better was Spock, and part of that was because of his appearance on Star Trek: The Next Generation.
Something was always missing when it came to Dr. McCoy, and, to this day, those empty holes haven't been filled. Perhaps that is something Star Trek: Strange New Worlds will do should they decide to introduce the character. Of course, finding someone to play it as well as Kelley did will be challenging, to say the least.