If your having to try that hard then yes something is probably wrong with the button, I have had no such issue and find both power and eject responsive to a simple touch. Yes there is and eject option from the home screen select options then eject disc.
So if the power goes out while in instant on mode will it cause damage? I wonder because the PS4 has issues when in standby mode if the power goes out. I'm getting a One soon and want to know if I should avoid instant in due to having a semi unreliable power grid.