The smallest email in my inbox right now has 913 bytes of headers and 132 bytes of content. Between the various hops, including through a spam filter, it took 13 seconds just to land on my server. It was then an additional 0 to 300 seconds before each of my devices knew it existed, because push remains black magic when dealing with clients and servers from different authors/companies.

This is not comparable to the normal performance of SMS or iMessages.