This is inaccurate. Horowitz is almost certainly using the term "socialism" in the sense that it was used by Hayek: i.e., as a state-managed economy. This is what he means when he writes of the "demise of really-existing socialism"; the idea of the state-managed economy is not an idea anyone takes seriously anymore. This is also what he means when he talks about how "the market had won"; the idea of a Soviet or Maoist-style planned economy has been completely discredited.
Socialism in that sense is completely top-down. The means of production is state-owned and wealth is distributed according to some model decided as best for the country.
But I take note of:
From the Old Right of the 1940s through the Reagan era, libertarianism’s opposition to socialism, especially interferences in the market, led us to ally with the forces of reaction. But even with the demise of really-existing socialism, we have been unable to completely break free of that connection to the right, though things are better than they used to be. . .
From this context he considers interference in the market to be socialism.
Nor is it true that classical liberals never accept state-intervention in the economy: classical liberalism isn't of necessity committed to laissez-faire; Hayek, for instance, is one of the more relatively statist classical liberals, as he openly denounced laissez-faire. Classical liberalism is a very diverse tradition, ranging from anarchists to figures who supported modest intervention in the economy, e.g.: Hayek, Mill, etc.
If we are talking of liberals, then we can be broad enough to accept this, however the defining feature which separates Social Liberals from Classical Liberals is support of laissez-faire capitalism. Classical Liberals support it, Social Liberals do not.
What Horowitz means when he writes that classical liberalism didn't figure out how to respond to socialism while maintaining its progressive credentials is this: classical liberalism is, in its values, a leftist movement. It stood for working class liberation and empowerment, tolerance and privacy in the private sphere, and so forth. The challenge from socialism was to say that markets don't empower or liberate, but rather oppress, and that a managed economy is necessary for social justice. In opposing this view, classical liberals in the 20th century found themselves on the same side as their traditional enemies (conservatives).
If you put it this way, then yes, I agree. However this is classical liberalism, not libertarianism (which is actually conservative, class-based ideology in a wrapper of Classical Liberalism).
With the demise of socialism, Horowitz and other left-libertarians believe that it's time for libertarianism to reclaim its roots of standing up for the empowerment of labor through traditional liberal means (i.e., ending pervasive government interference which favors big business), and once again championing the idea that true free markets (not, mind you, the corporatism that conservatives try to pass off as "free markets") are the best way to accomplish left-wing goals.
This is the primary statement I disagree with. Libertarianism was NEVER about empowerment of labor. Classical Liberalism was, social liberalism is. Libertarianism is an ideology which draws from the notion that those who are well-off deserve to be (this is axiomatic), and that they do not owe society for their success and neither do they bear responsibility to society (again an axiom, it cannot be reduced further, and it is a value statement which means if you disagree with it, you oppose libertarianism and will never see eye-to-eye with its advocates). Libertarianism takes this notion (which is deeply tied to property as a measure of being both well-off and free) and then borrows Classical Liberal positions on drugs, sex, and social freedoms.
Note your qualifier "true" free markets. Libertarians MUST use qualifiers like these because history is full of examples of unregulated, open to competition, virtually anarchistic markets which became travesties of human rights violations and social menaces. If you did not use the qualifier "true" then I could simply list these examples and the debate would be over, but for every example I list you need only say "well, that wasn't a true free market" and then your hands are washed of it.