You have to understand that concepts like rotational kinetic energy are just shortcuts to solve problems efficiently.
In Classical Mechanics, we start by defining concepts like kinetic energy on point particles, with no dimension; when we need to extend this concepts to macroscopic objects, like the rod in your problem, the rigorous way to do it is to think of the object as a collection of point-like sections, each having its own kinetic energy; and then we can state that the sum of all those point contributions is the total kinetic energy of the object. But since the sections are really small and really numerous we have to perform an integral instead of a sum.
This is what is truly happening at a rigorous level, but of course no one wants to do all this complex work for a simple problem, especially in a introductory context. So we define things like the kinetic energy of the center of mass, or the rotational kinetic energy, to skip some work.
In your case you can choose what approach you want to use:
- You can perform the integral (I do not recommend this one of course)
- You can think of the rod as a point particle and calculate the kinetic energy from this prospective. (So using the kinetic energy of the center of mass)
- You can think of the rod as a macroscopic object and calculate his rotational kinetic energy, that would be also the total kinetic energy.(Of course the pivot has to be the fixed point)
All those different approaches will give you the same answer for the total kinetic energy, as you can test for yourself. Of course this is not a coincidence: for example to calculate the rotational kinetic energy you need the moment of inertia of the object, usually at an high school level this quantity is given, but if you want to calculate it yourself you have to perform the integral that we were talking about before! So you can see why the first and the third method must agree.
The second method works thanks to the theorems regarding the center of mass in Classical Mechanics, long story short: in many situation you can approximate a macroscopic object with his center of mass with impunity, and this is one of these cases.
: Keep in mind that the moment of inertia of a rod rotating about an axis passing by the center of mass is not the same as the moment of inertia for a rotation about an axis passing by one of the ends, which is your case here.