Change one major historical fact and lots of things change. The Republicans can become stronger. The US changes from a non militarized county to militarized country. Prussia/Germany grabs the chance to unify faster and further militarize. The 1870-71 war begins sooner. Prussia sees the chance to humiliate France while Russia sits it out.
But these things have to result from actual drivers.
For example, going down the list here:
In the event of British intervention and a significant defeat for the US, the Democrats are going to paint it as a Republican mistake. This is the kind of thing they have good reason to do and it's the kind of thing that frames the disaster in such a way as to give them an electoral boost.
This has to be borne in mind; there is going to be a significant faction in the US which is blaming everything that just happened
on the Republicans.
The US changing from a non-militarized country to a militarized country is quite plausible. This will have knock-on effects on the US economy (which will contract) and immigration (which will also contract).
Prussia taking the chance to unify faster is... less plausible, because the "chance" does not really exist. The reforms which made the Prussian army one which could take on and defeat France took most of the decade from 1861 to 1871, and the cooperation of the German Federation was secured as a result of several years of setup plus a war in 1866; at the same time, there was no real risk of Russia intervening in a French-Prussian war in the 1860s, so Russia "sitting it out" doesn't actually gain anything for the Prussians. There is however significant unrest brewing in Congress Poland.
In 1862 the Prussian king nearly abdicates.
You might not see the threat of Confederate conquest, but you are not a mid 19th century American. And you ignore the fact that many northern Democrats felt personally betrayed by the secessionists.
But you're going from "mid 19th century Americans felt this was plausible" to "it would happen". There's not much indication that the British, for example, felt it was at all plausible, and it's
that which would drive British decision making; there's not much indication it would actually
happen, and it's that which would drive future events.
We have so far supposed that the British follow recognition with breaking the blockade, but there is no reason to assume that. The British might still have enforced most of the requirements of neutrality.
Change one thing and many things change. And as far as not making sense, it does not make a lot of sense that a Corsican with Italian relatives would rise to become the autocratic ruler of France, but he did.
I'm not sure you're actually reading the many caveats I place on my posts... I have more than once said that a war developing is distinct from the question of recognition, but that exploring the
consequences of such a war helps to explain why historical actors would act the way they do.
As for making sense versus not, that argument doesn't really wash. It is arguing that your assertions don't have to make sense because real history doesn't make sense, but we must have some basis for analysis of a historical or alternate historical question and "makes basic sense" is the only starting point we can apply.
The idea of a strong man rising to become the ruler of France is a possible outcome of the chaos following the French Revolution, and where he came from isn't really relevant to the broad-strokes analysis. A Napoleon-alike of similar skill who was a Belgian or Parisian would have had fundamentally similar impacts on history, though it is worth realizing that another plausible outcome of the events post-Revolution is the establishment of a French republic which actually survives and lasts.