If it hadn't been aes-ni, it would have been chacha, which is much faster than unaccelerated aes and close to the speed of accelerated aes.

Phones use https without a problem, and those haven't had hw-accelerated aes until recently.

A phone needing to set up a dozen HTTPS sockets is nothing for the CPU to do even without acceleration. A server needing to consistently set up hundreds of HTTPS sockets is where AES-NI and other accelerated crypto instructions becomes useful.