I found what Semaphoric said to be true, if I drag a guide too fast it might overshoot the center snap zone before it can snap, but if I drag slower I can see the displayed position value stop and hold near the center as it snaps.
View > Snap is a master toggle switch for all selected items on the View > Snap To submenu. For example, if only three of the options on View > Snap To are selected, View > Snap toggles those three on or off together. In other words, Snap To controls what get snapped, and Snap toggles what you set up in Snap To.
One thing that can affect snap to center is that if a lot of Snap To options are on and layers already exist, a guide might snap to layer edges that are close to the center, making it difficult to lock on the center itself.
In order for the guide to automatically snap to the center, should I have been using the Snap To > Guide?
No, that controls whether a layer you drag snaps to a guide. It doesn’t seem to affect dragging a guide, since it would mean snapping a guide to another guide which would put them on top of each other.