More specifically, is a Gameplay Designer an entry-level position? If not, what is a typical path someone would take to become one? Is there only one path, or are there multiple different entry positions someone could take before ending up as a Gameplay Designer?

Ton Sloper usually starts posting links to various articles that he wrote about the topic, https://sloperama.com/advice/designprep.html​ looks like a useful one, but there are more articles about the topic, you may want browse the collection.

Alberth said:
Tom Sloper usually starts posting links to various articles that he wrote about the topic

Thanks, Alberth. Here's another link, apropos to “path toward becoming a Gameplay Designer.”

Playing the Upgrade-to-Designer Game

Basically, @devbot , get any job in games. The more experience you have in the industry, and the more practice you have designing games, the better your chances of leveling up to Game Designer.

Hope you're not a bot, btw.