Secondary question would be were building heights subsequently higher as elevators became more popular and was it in any way connected to elevator engineering?
Until the 20th century, the tallest buildings were churches, which had only very limited public access to upper levels. You can see the listings here.
The safety elevator (invented 1852) was one of a series of developments—including steel framing, plate glass windows, and telephones—that made tall office buildings practical. These various developments, and the concentrated demand to support building projects making use of them, came together in New York and Chicago in the 1880s.