In a Yagi, the beam direction is the result of constructive interference between all three elements. Typically the elements are spaced about a 1/4 wavelength. The reflector is ahead about 90 degrees (a 1/4 oscillation) in phase, and the director behind about 90 degrees. The phase of each element is a consequence of its spacing and length.
When viewed from the front of the beam where gain is greatest, the reflector is a 1/4 wavelength farther away, and the director a 1/4 wavelength closer. This counteracts the phase shift just described, and the result is the wavefronts from each element are arriving in phase from this perspective. The constructive interference is what gives the antenna high gain in this direction.
Behind the beam, this is reversed: the relative distances of the elements adds another 90 degree phase shift on top of the 90 degree shift they already had, meaning each individual wavefront arrives 180 degrees out of phase. These opposite waves cancel, meaning the antenna has very low gain (ideally zero gain) in this direction.
By Chetvorno [CC0], from Wikimedia Commons
The reason for the radiation pattern of a dipole over ground is similar: it's the consequence of the wavefronts from two radiating elements arriving in phase or out of phase. Only in this case the 2nd element is the image antenna that exists as a consequence of the ground. As a simplification, whenever an antenna exists on one side of a conductive plane (like the ground), we can imagine that plane as a mirror, and pretend there's an identical antenna on the other side of the mirror. The only difference is the phase of this image antenna is opposite.
So consider a dipole a quarter wavelength above ground. This makes the image antenna a half wavelength away from the dipole, and 180 degrees out of phase.
At low elevation angles, the distance to the real dipole or the image antenna is equal, so there's no phase change. So the wavefronts arrive 180 degrees out of phase, and cancel. So no radiation here.
Directly overhead, the image antenna is a 1/2 wavelength away, adding 180 degrees of phase to it. Now the wavefronts from the real dipole and the image antenna arrive in phase, so there is high gain in this direction.
Raise the dipole to a half wavelength, and now the relative phase and distances of the image antenna result in a null directly overhead, and a maximum at 30 degrees above the horizon.
Now to your question: could adding more elements to a dipole that is less than a quarter wavelength high improve gain at low elevation angles?
Well, depending on just where they were added, and their relative phase to the main element, sure. However, since you're already starting from a situation where some of the elements aren't in the right phase, you'll never get better than a compromise solution. And at least some of these elements will need to be at least a half-wavelength high. If the support structure for holding an element that high exists, why not just put a simple dipole up there and skip the reflector business?
Furthermore, proximity to the ground has another detrimental effect: restive losses. The ground isn't a perfect conductor, so whenever there's current in the ground, transmitter power is being lost. This is why we put down radials for example. Getting the antenna higher reduces the current density in the ground, and thus the loss. To have some kind of antenna close to the ground which is still efficient will require improving the ground conductivity somehow.