This isn't the TOS Spock I remember. He starts out wearing his emotions much more readily. In "The Cage" he is more demonstrative than almost any of the other characters. He is more reserved by WNMHGB, but not much.
Keep in mind that the pilots--especially the 1st--were not designed to show deliberate or rich character growth, as by their nature, a pilot is an isolated piece, trying to sell a series. Nimoy has stated numerous times that the reason he started to alter his portrayal in WNMHGB (and beyond) was due to the personality and acting choices of Shatner, so there was no grand character design on the part of Roddenberry. The two actors found contrasts in the other they could work with--an artistic situation Nimoy did not have with Jeffrey Hunter.
Throughout the first half of that first season, he speaks loudly,
That was Nimoy still carrying over that--I guess you would call it--"military barking" from the pilots. But he did not behave that way throughout any of the entire episodes to follow, so it cannot be said to be some developing character trait in the way the open "dealing with my human side" would be later on. In other words, that "barking" was related to functioning on the job, as opposed to Spock's emotional state.
catches himself veering into human displays of emotion, in short seems less in control. If anything, I think he goes from being more comfortable displaying human emotions as a young man to vying within himself for greater and greater degrees of self control during TOS.
Spock was more than comfortable expressing whatever amount of human emotion for any situation throughout the series, which illustrates how much he had changed from trying to fight his human side, to accepting it. Someone on YouTube posted the following link, which showcases Spock's growing comfort with humor, irony and sarcasm from season 2:
https://www.youtube.com/watch?v=ldezDyDDhRg
He was not clumsily experimenting with emotion, or faking it, but living within his "natrual skin," which makes any post-TOS (TMP) idea of Spock trying to shed himself of all emotion completely out of character with no precedent, since TOS was not sketchy on how settled Spock was with successfully merging his "of two minds" state.
he realizes the Vulcan way cannot encompass enough of what he has experienced.
One would argue that his experiences in
"Amok Time" and his relationship to his mother (a window into that seen in
"Journey to Babel")were proof that he was covering that part of the journey.
Spock was Rodenberry's creation and he obviously thought this way too, because he chose to give Spock this story.
That does not mean his ideas were always best for the character--and he certainly was not the only person responsible for the development of the Spock that was so well explored on TOS.
And while other forces were also at work, this idea of Spock's evolution must have appealed enough to Nimoy to interest him because he, like Spock, abandoned his self imposed exile to return to the Enterprise.
Nimoy returning had many real world motives, too.