Why don't solar powered aircraft use wings with more surface area
First, because solar panels are pretty mediocre sources of power per area. An industry average found via quick Google search is in the 200 - 300 Watt per square meter range, so let us be very optimistic and take the higher value, $300W/m^2$, for a round of math. If you want the details, solar irradiation on the surface is about $1kW/m^2$ and the best experimental photo-voltaic cell efficiencies by year are graphed here.

Using the ASH 26 motor glider as a case study, we can see if solar powering an aircraft is as trivial as the OP suggests.

Assuming our panels are made of fairy dust we can further decide their weight is negligible and that we can perfectly cover the $11.68\,m^2$ of wing area this glider has and that it will operate as though under normal incidence (Sun at 90° to the panel) all the time. Thus, $3.5\,kW$ of free energy... which sadly is only $9.4\,\%$ of the power supplied by the Wankel engine used on the real glider, and with which it attains a very humble $4\,m/s$ climb rate. This is before even considering that batteries are also very inefficient sources of energy per unit of mass when compared to hydrocarbons.



Now, for the second part of the OP´s query, how about enlarging the chord of the wing to increase the area?
This also happens to fall flat for different reasons. The drag coefficient:
$$
C_D =C_{D0} + \frac{(C_L)^2}{\pi e AR}
$$
has the aspect ratio ($AR = {b^2 \over S}$) of the wing in the denominator of the second term, the so called induced drag (because it is induced by the lift, note the coefficient of lift itself appears in the numerator).

This equation quickly illustrates why gliders have long wings: a higher AR provides a lower induced drag term. The other term, the parasitic drag, is either indifferent to the chord or slowly grows with it due to boundary layer transition in laminar airfoil designs.

We can arrive at the conclusion that for a given surface, a higher-AR wing will be the less draggy solution.

Therefore, the current attempts at solar powered aircraft all attempt to add area by extending the wings, and are limited by structural constraints like bending moment at the root, which you can clearly see them try and alleviate by spacing out the engines into the wings, providing some relief at the expense of increased roll inertia.