I stumbled on the issue, in case others find themselves in a similar situation:
I had a layer with the Templater effect that was thrown outside its comp’s timeline. Whether this was a result of user error, or I had deleted a layer and forgot to update the effect settings, I am not sure.
Once I found the culprit layer, I deleted it, re-created the layer, and re-applying the Templater effect. This caused it to behave normally again.
I wish I knew the cause of the issue that made the layer appear outside of the comp’s timeline so I could avoid it in the future. But, for now, this gives me something to look out for in case the issue pops up again.