Why can't rotational energy and translational energy simply be added up in this problem?

This question comes from a previous homework problem that I already have the solution for but I’d like to gain a greater physical insight into the solution. In the problem below (problem 7-19 from Principles of Dynamics by Greenwood), a block is sliding along a frictionless surface. The motion is caused by a rod attached to the block falling due to gravity.

In the solution to the problem, Lagrangian mechanics are used, so the total kinetic energy (T) is summed as shown below. In the top equation, the first term is the kinetic energy of the block. The second term is the translational energy of the center of mass of the rod which is found using the schematic below and the law of cosines. The third term is the rotational energy of the rod taken about its center of mass.

This solution method makes perfect sense to me but when I first attempted to solve the problem my instincts were to take the rotational energy of the rod about the connection point using the parallel axis theorem then simply add the translational energy of the rod, which would be the same as the block since they are connected, to get the total energy of the rod (and add the kinetic energy of the block too). If you do this, you would get:

This is similar to the actual solution, but does not contain the cosine term, so the equations are only equal when theta equals 90 degrees. This second method of solving is clearly wrong, I’m not disputing that, but it seems like it would be a valid way of solving the problem. Does anyone have a sense of what the cosine term physically represents and why the second method inadequately does not include it? In other words, what is the physical meaning of the energy being left out by the second method.