Yagi Calculator vs. Bruteforced values?
Many times, gain calculations don't account for input impedance. Most conventional RF components are either 50 or 75 ohms in input impedance. To maximize power transfer between a component and a system (such as a 50 ohm cable and an antenna) the component impedance must equal the complex conjugate of the input impedance to the system.

The input impedance that you provide in the figures is 6.10 + j95.25 ohms. This is problematic for two reasons. First, the magnitude of the reflection coefficient as seen through a conventional 50 ohm system is 0.9488 (0.0512 transmission coefficient). This means that around 90% of power (the square of the reflection coefficient) that you put into the antenna through a 50 ohm antenna is wasted. 

Another issue is that the most of the input impedance associated with the antenna is imaginary. Imaginary components for antenna impedance correspond to power that is stored in the near field, which is undesirable for propagation over large distances relative to the wavelength and dimensions of the antenna. 

I do not know how the impedance from the steps provided by the program compares to the impedance of the antenna that you designed because that value is not provided. It could be the case that the antenna produced from the steps in the program results in a better impedance, and therefore less wasted power. Regardless, the fact remains that the impedance provided by the antenna that you designed is mismatched to most conventional RF systems.