As far as I can give an opinion on this you seem to be locked into the world of iOS. I looked up Fitbit, Amazon and eBay from your list and a lot of apps on my phone and all of them have the burger menu. Out of all posibilities it is THE icon every user of an android smartphone (around 80–90% of all smartphone users) will always recognize as the menu.
Twitter is the only exception from this listbut still has a menu hidden behind a swipe-in-from-the-right navigation, what in other apps is the same as the burger-menu. Even worse because you do not have a visual indicator of that menu. You never know if you do not try it out, and most users do not try out anything from my experience.
This beeing said there is an easy and good explanation of why so many people use the hamburger. It is well known by almost every smartphone user and they expect it to be there. Otherwise your users will be confused.
But a hamburger does not mean that your main activity can not be intuitive and lead the user through the app without it. Amazon does this quiet good, I never need the hamburger for buying something. But when I look for a page I can not access right from the current activity I as a user always know I can access everything from the hamburger.