Of late I am coming across situations where people are left with no choice but to install an app - even if they don't trust it. For example, I had resisted installing WhatsApp on my dad's phone, but had to give in when one of his clients requested him to send a document over WhatsApp!
Some e-commerce businesses are going app-only in India (Myntra for example). There are rumors that Flipkart might follow next. I love the Flipkart website itself and use it extensively. However, if they go app-only and the time comes for me to install their app - I might be cautious if the permissions are not to my liking.
Then there are taxi aggregators in India (Ola cabs is just one such example). Not only do they offer discounts if you use their app, chances of finding a cab are better if you use the app versus calling their toll-free number to book one. But then, there is again the sticky issue of permissions.
Point I am trying to make is - there are situations where you are forced to install an app that you might not entirely trust with all the permissions. The permission system is definitely a very big deal for such cases.
And we have not even talked about the non-techie folks who use Android and who blindly install apps that they want. The permission system is for their protection too.