I was actually going to say something about the ham technique being very stage actor-like, but hadn't known about this particular play. I could easily see this. Part of why the stage favors this style of acting is you have to project yourself. You don't really have machines doing the job for you. So exaggerated facial expressions, being loud, chewing scenery, all helps. Stewart also started out as a Shakespearean stage actor, and while he was somewhat similar in TNG, he was more restrained in his delivery of speeches.
It's somewhat ironic to think Shatner started out as a stage actor, while Leonard Nimoy started out as mainly playing thugs. James Doohan was on the Howdy Doody Show. You might have seen DeForrest Kelley in old Western shows and movies, like Bonanza and Gunfight at the O.K. Corral (for some reason he was in three different versions of this maybe just from being good at it).
He was far from the only Star Trek star who showed up on Westerns. Bonanza has a lot of them. Did you know Ricardo Montalban showed up on it? So did both Leonard Nimoy and DeForrest Kelley.