This doesn't make sense at the micro level though. You have (let's say) half a dozen oil companies all producing roughly x BBL for $y/BBL. What's to stop Company A, who has the means to do so already, from simply producing 1.1x BBL and keeping its price at $y?

At a macro level yes, if everyone increases supply 10% you'll see some price fluctuation, but that doesn't necessarily hold for an individual, self-interested company. Hell, that company could produce 10% more but sell it for $y-0.05 and presumably make more money selling for cheaper.