You are correct about overhead wires being better. They are better in EVERY way!
1. Higher voltage can be used.
a. Smaller size wire can be used.
b. Sub-stations can be fewer in number and farther apart.
c. More trains can be operated close together on the same sub-station.
d. Higher top speeds can be attained.
e. Acceleration is better, especially where trains are close together.
2. There are no flash-overs from accumulated wheel and ground dirt that third-rails
and their pickup shoes have.
3. There are no electrical hazards to persons on the ground.
4. There are no difficulties with track maintenance. Track maintenance is less
expensive.
5. There are no inherent restrictions on train speed. (Third rails have inherent
limits on train speed.)
6. There are no prohibitions on street-running or grade crossings. (This is not to
say that street-running or grade crossings are desirable.)
7. Power to cars is always continuous. Cars cannot get "gapped." (This is where
a car is stuck (stalled)in a gap between third rail sections and cannot
get power until it is pushed away from the gap by something else.)

Now, if anyone wants to argue the advantages of third rail, I would like to hear them.