After reading your post, I was curious how the chicken/egg problem was solved at that time. Searching for "first cars gas stations" on Google, I found https://en.wikipedia.org/wiki/Filling_station which states
"The first drive-in filling station [...] opened to the motoring public [...] on December 1, 1913 [...]. Prior to this, automobile drivers pulled into almost any general or hardware store, or even blacksmith shops in order to fill up their tanks."
So gas stations have actually not been built in a large upfront effort, but existing infrastructure was used to distribute gas.
"The first drive-in filling station [...] opened to the motoring public [...] on December 1, 1913 [...]. Prior to this, automobile drivers pulled into almost any general or hardware store, or even blacksmith shops in order to fill up their tanks."
So gas stations have actually not been built in a large upfront effort, but existing infrastructure was used to distribute gas.