A wider wire, let's say really wide (4/0) is being hit with more energy than a thin wire, is it not? A square foot of anything is exposed to one half the sunlight/energy that a two square foot anything is, when both are in sunlight.
Sunlight has extremely small wavelengths, several orders of magnitude less than (to use the distance you mentioned) a foot. This means that the interactions of any given portion of the light with a solar panel (or even an ordinary dark object getting heated) occur in very small areas of the material, so if you analyze it as if it were an antenna, it's more like an array of tiny antennas than a bigger-in-some-dimension single antenna.
In the case of a radio wave interacting with a dipole antenna, for example (not that far from the case of your quagi), the entire antenna is built around half-wavelength dimensions. The area of the incoming electromagnetic wavefront that this antenna is interacting with is not the length and thickness of the wire, it is more like a circle whose diameter is the antenna length. (Reality is much more complex than this; there are no simple geometric shapes, but this will do for now.) Thus, making the wire thicker does not gather significantly more energy.
The analogy to making your solar panel two square feet instead of one is making two identical antennas, not too close to each other and connecting them in parallel. This is not a perfect analogy, because solar cells (and heating) are treating the radiation as non-coherent, meaning the phase of the signal is discarded and only the amplitude is kept. For coherent reception, as used in radio communications, the math works out that any such array of antennas or any other means of gathering more energy makes the antenna more directional. That's why we measure both properties as a single value, the “gain” of the antenna.
You can in theory gather more energy from a radio signal without increasing directionality, by using multiple antennas and multiple receivers as well — by independently demodulating from each antenna you avoid the signals cancelling each other. Imagine building lots of crystal radios and combining all their weak audio outputs into one stronger one. When the goal is actually to gather electric power from the RF field (in which case all you need is the diode rectification, no filtering or audio transducer), this is called a rectenna and it can be built into an arbitrarily large array just like a solar panel.