As of last January, WhatsApp handled ~30b messages a day [0]. As of this January, WhatsApp has 1b users [1]. iMessage has ~300m and "several billion" messages per day (hard to get stats unfortunately).
Telegram has 100m users and handles ~15b messages per day [2] but, similar to Google, is not e2e encrypted.
As of last month [3], Facebook Messenger and WhatsApp combined had 60b messages per day. Assuming WhatsApp daily messages grew roughly linearly with users, their increase from 700m to 1b users would put them at ~42b out of that 60, leaving 18b for Facebook.
So there are at least ~40b e2e encrypted messages flying around thanks to WhatsApp, with likely at least 20b (assuming iMessage is bigger than Telegram, which seems reasonable) more from iMessage, putting the e2e encrypted count around the ~60b ballpark.
Telegram and Facebook together would be ~30b and I seriously doubt Skype IM is competitive but KakaoTalk is ~150m users, around the same size as Telegram and WeChat is around 600m users [5].
So the unencrypted messages definitely have far more volume and users than encrypted but the encrypted platforms are still the largest around. The article said "all of its biggest competitors" and given that Google doesn't operate in China (ruling out WeChat), I think that's a reasonable choice of words, although it's really hard to say thanks to iMessage's lack of transparency.
[0]: http://venturebeat.com/2015/01/06/whatsapp-now-has-700m-user...
[1]: http://www.statista.com/statistics/260819/number-of-monthly-...
[2]: https://telegram.org/blog/100-million
[3]: http://www.theverge.com/2016/4/12/11415198/facebook-messenge...
[4]: http://www.statista.com/statistics/248863/number-of-register...
[5]: http://www.statista.com/statistics/255778/number-of-active-w...