That fear disappeared with the East Texas oilfield discovery in 1930. Which so increased the supply of oil relative to demand that prices fell to 13 cents per barrel.

This created a number of problems, including the prospect of damaging oilfields to the point that future extraction would be compromised. It ended up with the governors of both Texas and Oklahoma calling out the state militia, and, in Texas's case, the Texas Rangers, and seizing control of wellheads by force of arms in an effort to constrain extraction and drive oil prices up -- to $1/bbl.

This resulted in the Texas Railroad Commission effectively controlling US oil output (with oversight from the US Department of Interior) from 1931 to 1972, at which point, peak US oil meant that there was no longer any surplus extraction capacity to limit. Shortly afterward the Arabs tried another of their periodic embargos against the US and Europe, and, to everyone's shock, it actually worked.

If you look at the price of oil, from 1931 to 1972 it was remarkably stable. Even WWII and the post-war consumption boom barely moved the needle. Post 1974, everything goes all to hell. We're still there now.

Daniel Yergin's masterpiece work, The Prize, covers this history in great depth.