I'm not really sure that holds water. For one thing, a central currency was going to be a thing, and that it had to be backed by specie was something the Hamiltonians and Jeffersonians and pretty much everyone else actually agreed upon. For another, the nationalistic military spirit faded out quickly in OTL (and was particularly done away with after the Jeffersonian 'revolution of 1800'). This obviously had negative effects regarding the USA's military capabilities (and particularly the navy, since a standing army was largely opposed anyway... at least by everyone who wasn't a very convinced Hamiltonian). Yet... it didn't lead to the USA getting "crushed by Britain". We saw what happened in the War of 1812: Britain wins. and then... does nothing. Because at that point it was absolutely clear to britain that it wasn't realistic to get the USA back into the British Empire.
I don't think it needs to be "small", either, although probably smaller than in OTL. Population drives are still going to lead to westward settlement, in areas Spain (and later, Mexico) cannot effectively project power. A lot of OTL's Louisiana Territory is probably still going to be settled by people coming from the USA. And if the USA is a highly decentralised confederation... there's no point in being opposed to joining it. Access to the internal market can only benefit the western settlers. On the other hand: I don't see a decentralised confederation fighting the Mexican-American War, or successfully challenging Britain over Oregon Country. So... no west coast, probably.
That leaves us with a smaller (but by no means "small") USA. As for it being "weak". Well, what's weak? It's central government would be extremely weak. But local governments would probably pick up the slack in many cases. I've often compared this to the EU: in a scenario like this, US states will act a lot like EU member states. It's not as if because things don't get handled at a federal level, they don't get handled
at all. I don't think this USA would be "weak". Certainly unable and unwilling to become a world-dominating superpower, but not some pushover that can just be conquered. (Maybe by a great power like the British Empire, but then they'd just have to deal with constant rebellion... so they wouldn't
want to, just as they didn't in OTL.)
Finally, I'm not at all sure this USA would be egalitarian. It would probably vary on a state to state basis. Since there is no central authority setting the tone on this, and states are more or less sovereign entities united in a confederal leage, some of the states will be very egalitarian. Others might go in the exact opposite direction. (I'm thinking of the Deep South here, with it's planter aristocracy, its slave underclass, and its negligable white middle class. But I'm also thinking of a potentially very Hamiltonian northeast, where wealthy financiers and industrialists would probably get to control the government, and a typically Hamiltonian disdain for "mob rule" would likely hamper democracy.)
To bring this back to the subject of the OP posted by
@Skeletor 1917 : in such a highly diverse union of states, there would be far more space for interesting socio-political experimentation. The USA invited a lot of that in OTL, even. People to, ah, "interesting" for European governments' liking had a tendency to seek out the USA, where they could explore certain radical ideas. This alt-USA would be even more fertile soil for such things. We might see (to stick to various anarchist-leaning examples) analogues to David Henry Thoreau, Josiah Warren, Lysander Spooner etc. gain more traction.
However... the idea of a "second revolution" resulting in something akin to anarcho-capitalism seems highly implausible to me. Most anarchists in OTL (regardless of the
kind of anarchist in question) were reacting to what they perceived as the oppressive nature of the state's power. In a highly decentralised USA where some states are really hand-off when it comes to governing... well, those exact states will attract the kind of people opposed to such uses of state oppression. And since they'll already be in a pretty free, unbothered situation, they'll actually be less inclined to go and overtrow what minimalist government there is. Conversely, any more authorative state government would draw the ire of (proto-)anarchists, but rather than overthrow that government, I think they'd be more inclined to just pack their bags and move to the most laissez-faire of the small government states they can find. So, irony of ironies: a realistic situation that is
closer to what a typical anarchist probably desires will actually make an anarchist revolution very unlikely!
That said, I can offer you some possible avenues of development. For one, if it is indeed the case that there will be various small government US states in this ATL, and if those indeed attract a high(er) percentage of people inclined to an anarchist mindset (or at least a leave-me-be-in-peace libertarian mindset), then this tendency in migration preferences will mean that such states will gain settlers who like it there, who will endeavour to
keep that state's government small.
Now, to consider which state(s) that might be: I've argued that the northeast is not likely for this (barring, say, the rural hinterland of Maine, New Hampshire and Vermont, which was pretty much out of government's reach
de facto for a long time in OTL). Nor is the deep south, with its planter aristocracy (who will keep government small as in OTL, proobably, but not out of any love for anarchist experiments). I think the most likely area to attract people inclined to dislike too much control over their lives is, logically, the frontier. The west. The Appalachians and beyond.
If that is indeed the case, then the factor of "the right kind of settlers" being attracted is even more important: such places would not yet be settled (...by white people...), they's be geographically most distant from any existing government, and if the bulk of the migrants actively desires a life free from too much government, you've got the basis for a pretty libertarian-ish region right there. Since the more "governed" states back east would not be attractive to anarchist-leaning individuals, these would keep being inclined to go west, to (say) "the Trans-Appalachian Free State". Thus giving this region a steady inflow of people likely to think and feel that way, which would enforce its cultural identity as being very much libertarian, at least.
Of course, libertarian =/= anarchist. But just as the central government not doing things often means that the local government will pick up the slack, a situation where local government is
also purposely kept very minimalist will probably mean that "the slack" is picked up by non-govermental parties. I mean, when I imagine a hypothetical more-or-less libertarian community, I don't really expect to see the oft-projected horror show of a country where people die in the street because there's no goverment-organised pensions or health insurance. People
want such things, and if governments don't provide them, others will. So in the minimal-government "Trans-Appalachian Free State", one might see a proliferation of community-based mutual funds to organise health insurance. And one might argue that such non-govermental solutions are
by definition anarchist solutions.
All in all, I wouldn't be suprised if a highly decentralised USA would end up with at least one rather libertarian state/region, which would in practice be very anarchistic (all though not explicitly or fully anarchist) in nature. There would be plenty of states/regions going in completely different directions, of course. That's rather the defining feature of such a decentralised system. So a fully anarchist USA (be it anarcho-capitalist or otherwise) still seems almost impossible to achieve. Unless, of course, the other regions get tired of the decentralised sytem, and secede from the USA to form their own (more centralised) unions, leaving the Trans-Appalachian region behind as "the USA"... then you'd technically get a more-or-less anarchist USA.