At my company it’s lateral.

Lead -> EM

Sr. Lead -> Sr. EM

Principal -> Director

Sr. Principal -> Sr. Director

The pay is aligned with the level whether or not you’re a people leader. To your point though, it may be difficult to go from Principal to Director. I see the lateral moves happen more at the Lead/Sr. Lead levels. They might do a Principal to a Sr. Manager as a trial period with the expectation that you’d be Director after a short time if you perform well. I’ve definitely seen directors become principals as well, so it goes both ways.