This is what I can see by clicking on the page you provided.
In my opinion the description is far too vague for it to be considered a complete breakdown of Hertz's offering.

Am I correct in understanding you that even if I do sign up for the $7.99/day premium assistance, I will still not be covered in the event of a flat/punctured tire? I am expecting the basic service of towing the car and tire replacement if such incident does occur.

Tire problems are far more likely to happen than losing/locking keys, running out of fuel,etc..