At least on recent Android versions the apps only ask for permissions when they actually need them, not on install time, and you can grant/refuse individual permissions.
In practice, apps that want access to my contacts usually keep asking for access every time I try to do anything, until I eventually either relent to make the prompts go away or click allow by accident.
And that's me as a computer-enthusiast. I would bet money most normal people just hit allow always. Because it's easier.
Some apps even try to circumvent this system by showing a "help" screen with instructions on how to re-enable the permission manually, but I only saw this twice.
Still, this is not practical enough. A true user-centric strategy would be to offer "mock-permissions" to an app, so that if an app says that it needs to read your home dir, you grant mock permission to the app, and it sees an empty home dir, not yours. From the point of view of the app, it should be impossible to know if it has been granted the real permission or just a mock permission.