The capacitance is directly proportional to the area of the cross
section. In transmission lines area is the product of thickness of the
conductor and the length. As we know the thickness is going to be few
centimeters. Hence the capacitance mainly depends on the lenght of the
transmission line. Hence only if the length is large the capacitance
value will be cosiderable. In short transmission line length is small,
hence the capacitance effect is negligible.
While the total capacitance of a transmission line is directly proportional
to length, the capacitance per unit length is not directly proportional to
the conductor cross-section (In addition, the effects of this capacitance is
not directly proportional to length). I suggest that you look up the
relationships dealing with capacitance per unit length for a transmission
line. Then look up the "telegrapher's equations". Your model has problems.
However, beyond these academic quibbles, I do agree with Tim Perry and you
that for a short line, the capacitance is negligable.