This engineer was in the perfect environment for someone who wants to grab more responsibility: "My boss [..] pulled me aside one day. He explained our team was growing fast and it was getting tough for him to have 15+ engineers reporting to him directly."

I think this sort of opportunity is more common in small, rapidly growing companies. Something (customer growth, funding, goals) is increasing work faster than hiring can keep up, so insiders are eagerly tapped to take on more. Such environments also have downsides.