Maybe you already know, but you can access the screen you mention by pressing and holding the home button.
I do think that this is the classic Nexus way of using the buttons, which indeed is widely used in Android world. The FP being my first smart phone, I would be bewildered, if the buttons would change their functionality to something more “nexus-like”; I would also need some time to get acustomed to another smart phone. Try to be a bit patient with how the FP works! 
And if you really want to change the button’s functionality I am sure, there are apps (Gravity box maybe?), which can help you achieve this goal.
If you don’t like the quick access, which is part of Fairphone OS launcher, just install a different launcher. You can find some examples here: How are your home screens set up?