I still don't understand how weakening a potential future ally is to their advantage. And considering that is the OP's question, not the feasibility argument we've spent so much time on, this premise demands more explicit examination.
I think this view suffers too much from presentism to be honest. We see the 20th century through the lens of the Great War and all of its consequences, and many people look at the events from almost 1814-1914 as a continuous build up to that great calamity which lead to the other calamities of the 20th century. But let's look at things on the ground in 1860 from both a British and American perspective:
From the British perspective, the United States is a rising power who they have fought wars with or been on the brink of conflict with since its inception in 1775. Partially because of British intransigence, but also because both parties had overlapping goals which did not align with the others. To the British of the 1860s, the United States is gobbling its way across the continent, is sometimes loudly declaring its intention to take Canada by force, and is potentially destabilizing influence on Britain's world trade network. To make matters worse, they are sympathetic to Britain's major rival Russia.
In 1860, the British weren't really afraid of the United States (small navy, small army, ocean away) but they definitely saw them as a rival, not a future ally. The United States meanwhile, was at least aware it had things to be worried about from the maritime powers of Europe like France and Britain.
From the American perspective, Britain is still perfidious Albion, a symbol which politicians will wave as a bloody shirt to whip up the crowd (see William Seward or 34/40 or fight!) which still gets dunked on in popular media. They are also a major trade partner, but not seen as a trusted friend. They also remember things like the attempts of British politicos to enlist Americans for service in the Crimean War, the Oregon boundary dispute and the most recent but absolutely ridiculous abortive Pig War. They also object to British warships seizing and searching American vessels off the coasts of Africa who are suspected in engaging in the illegal slave trade. There were still many reasons for them not to trust the British politically, but numerous others for them just to not interest themselves in European affairs.
Neither country in 1860 would have looked favorably on the idea of an alliance. They both would have thought there was no reason for it, too many strings attached if it came, and it would run contrary to any idea of independent foreign action they might have.
In a situation where Britain did tick off the Americans in this fashion, the British might just sit in their bubble of splendid isolation if similar circumstances brought up an alt Great War. Their fleet will still probably be powerful enough that each alternate alliance would have to give pause before inviting it to join any other side, and their economy would still be one of the largest in the world. They could probably quite comfortably sit back and play kingmaker from a diplomatic and military position.