The problem here is as follows: in the relevant time period, the state was pretty much a small bureaucracy, typically serving (to a great extent, at least) the interests of the elite. Historical revisionism notwithstanding, the conservatives often wanted it that way, while the (classical) liberals - even though they wanted a generally small government, a free market etc. - were in fact very often the people driving for reforms. I'm not exactly sure about America, but in various European countries, the (classical) liberals secured the passage of laws ensuring state pensions, government financing of education etc. etc.
Basically... the classical liberals and the 'moderate liberals' (who were precursors to... social democrats) were working hand in hand... to make the state (moderately!) bigger, actually. Conversely, lots of socialists initially saw the state as the instrument of the 'capitalists' (meaning, in their discourse, not 'free marketeers' but 'the really rich elite'). They were not exactly wrong. Very often, government forces were used to - for instance - violently break labour strikes. (And again... liberals typically voted in favour of labour laws! Albeit with reservations: their goal was limited state involvement, and certainly nothing like a social welfare state.)
What I'm trying to say is: given that situation, it is logical for socialists to be anti-state, and for classical liberals to be pro-state (or pro... a-bit-more-state). There's a reason even those socialists wh believed in a role for the state called for revolution: they saw the existing state as an enemy, to be overthrown and replaced. This is not to say there were no liberal thinkers who wanted something much like anarchism. But these were typically radicals who had at least some ties to the more "left wing" (proto-)anarchists.
If you want anarchism to arise from a classical liberal tradition, you need to make it so that the state is entirely in the hands of anti-market conservatives. This might prompt classical liberals to perceive state power as an intrinsic threat to human freedom, which would lead them to, well... something a bit like anarcho-capitalism, I'd say. (Although that movement is modern, and has its own quirks. Perhaps it would be more accurate to say that classical liberalism would turn into a movement that closely resembles the OTL ideas of Frédéric Bastiat.)
