Huh? Neoliberalism started as the rejection of the pure laissez-faire policies of Mises. It promotes state intervention in the markets.
I think you are confusing the mid-20th-century (1930s-1960s) use of "neoliberal" with the late-20th-century-to-current (1980s-and-later) use. The term basically fell out of use and was resurrected with a completely different meaning.
It was associated more with Pinochet in Chile and the US policy preference which favored Pinochet-like figures.
(Clinton in the 1990s saw the adoption of the new term "Third Way" for something which was somewhat similar to the mid-20th-century use of "neoliberal" -- and quite distinct from the then-recent resurrection of the term -- though the contemporary left-wing social and right-wing economic positions it was synthesizing -- particularly the left-wing social positions -- were somewhat different than those of mid-20th Century neoliberals; though there is a connection to the 1980s definition of neoliberalism, since the then-current right-wing economic position that was key in the 1990s Third Way was the new, 1980s-definition, neoliberalism. But 1990s [and later] Third Way-ism isn't the defining instance of 1980s-and-later neoliberalism, its just the most politically left-wing movement incorporating some elements of 1980s-and-later neoliberalism.)
> Only in a very narrow, utilitarian sense.
I think this is the core of our disagreement. Economy/market as tool is a principal tenet of neoliberalism, IMO.
No, your "oversimplification" is an overcomplexification: neoliberalism has nothing to do with the social left/right axis.
> and it's also the philosophy that the economy/market is a tool to be used to achieve goals.
Only in a very narrow, utilitarian sense.
> Neoliberalism doesn't specify what those goals are.
Actually, it does; neoliberalism rests on the theoretical optimality of ideal markets in maximizing experienced utility.
> That gets associated with the social left because the social right's goals are basically "keep the world the way it is and/or roll back changes made".
Insofar as that is an accurate description of the social right's goals, its because the social right consists of people who believe that rolling back (negative) changes is a way to "make the world better", often (though not always) in utilitarian terms.