It's not like Marianne Moore is giving us a guided tour through a museum or a lecture on ancient civilizations in "The Mind Is an Enchanting Thing." But she is showing us, in a subtle way, how the mind is responsible for art and culture. She mentions two bigwigs of classical music right away in the first stanza and comes back to one of them at the end. She even closes the poem with an important ancient figure. So even though art and culture aren't the focus of this poem, they're a byproduct of it, just like sweat is a byproduct of a basketball game. It might not be the reason you wanted to play in the first place, but you're going to get it anyway.
Moore chose to use famous musicians in a poem about the human mind because composing playing music takes a lot of creativity and brainpower.
Gieseking, Scarlatti, and Herod aren't specifically important in this poem. They're there as symbols to show us what great minds are capable of (or, in Herod's case, not capable of).