"I always thought you should basically have your antenna feed point be
directly over (or embedded in for through hole) a ground plane"
This is only true for some antennas.
Most generally: Try to keep the antenna as far away as possible from any electrically conducting materials, especially from metal surfaces.
Exception: With each antenna comes a specific field configuration (E-field & H-field). Metal surfaces are fine as long as they are strictly perpendicular to the E-field. The problem with conductive surfaces is that they short-circuit the E-field (force it to 0). As long as the E-field hits the surface strictly perpendicular, the surface is equipotential with respect to the E-field, and the field configuration remains undisturbed.
The exception is most commonly met whenever there is a symmetric property to your antenna. E.g. a complete di-pole has two axes, feed-point in the middle. In the plane perpendicular to the di-pole, right at the feed-point, the E-field happens to be perpendicular to the plane. You can thus replace one axis of the di-pole by a "ground plane", feed-point exactly where the now mono-pole hits the ground plane. This is also happens to be true for some other commonly used antennas.
On the other hand, you can use the effect as part of the antenna design in order to force the E-field into some configuration. This is done e.g. in some directional antennas.
Near-field vs. Far-field: The field of an antenna can be categorized into near-field and far-field. Field-disturbances in the near field are generally catastrophic with respect to the intended antenna performance, field disturbances in the far-field only affect performance in the direction of the disturbance. As to where the near-field ends and the far-field starts is non-obvious: Some antennas are more sensitive than others. As a rule of thumb: Everything 3-5 lambdas away is definitly far-field. Anything closer may or may not interfere with antenna characteristics, modifying its center frequency, directivity, matching, ...
The concrete antenna you are referring to has a helical shape. This thesis on helical antennas aproches helical antennas using two models:
- folded di-pole (circumference << wavelength): behaves roughly like a di-pole
- axially radiating helical antenna (circumference ≈ wavelength)
Judging from the radiation diagram, the antenna under consideration is somewhere between those two extremes, at least when mounted perpendicular to the ground plane. In this case the E-field is strictly perpendicular to the ground-plane. The feed-point should be right on the ground plane and the ground plane should optimally extend some centimeters in all directions around the feed-point.
If the antenna is mounted parallel to the ground plane, it will short-circuit the E-filed. The ground plane will profoundly change the near-field configuration and you therefore need to consider it as part of the antenna configuration. In effect, you are now looking at a totally different antenna, which is why the theory in the linked thesis does not apply any more. I bet the antenna will also induce a fair level of HF into the ground plane (normally considered problematic). As you might see from the radiation diagram, the new antenna is also quite directional with practically zero radiation in direction of the ground plane.
I have no idea why it is advantageous to keep a minimal distance between antenna and ground plane. Maybe to contain losses in the ground-plane, but could just as well be due to matching or tuning or directivity or all combined.