I think that’s a complete pipe dream.
First, just because it’s a more Northern city doesn’t make it that much less susceptible to summer heat; especially when we’re talking about a large city. Fetid, fuming, filthy Philadelphia wasn’t any more amenable to the Congress than bucolic Washington- why would New York City be a summer paradise?
Second, you’re assuming that being involved with a major metropolitan area rather than a backwoods town would have A) enlighted conservative politicians, rather than simply re-affirming their beliefs that cities were horrible, horrible places and B) that said leaders would have effected a vast sea-change in perceptions amongst the populace in general. Hamilton’s theories on government were not ignored in the early 19th century simply because American politicians were rural and interested in the agrarian ideal; they were ignored because the overwhelming majority of Americans were rural and interested in agrarian ideals and elected politicians who shared their views. Cart here, horse there.
Having said that, I do think there would have been major ways in which the course of history would have changed, but I’m not quite sure in which direction- could have gone either way.
First, and most obviously, the American Civil War. I think Little Nemo may be right- if the capitol were obviously northern, anti-slavery moves by the government would have been much easier to interpret as Northern aggression against the South, possibly leading to an earlier Civil War. On the other hand, with the city of Washington, NY not being a place where open-market slave auctions were held, it’s quite possible that some of the smaller issues that led to fighting would not have come up in the first place.
Had the war occured at the normal time, I also think it’s likely Maryland would have seceded. Lincoln’s actions to keep Maryland in the Union- which were much of the unconstitutional actions he took- were much easier to accept and ignore when the alternative was having the Capitol stuck deep inside enemy territory. With Washington, NY firmly in Unionist territory, Lincoln may never have taken those actions, or may have deemed the larger outcry not worth it.
Losing Maryland to the South (and likely Delaware, given its slave-holding and the fact that with Maryland in the Confederacy it would be in far too precarious a position to stay Union easily- see West Virginia for the opposite example) would have given both sides an advantage, paradoxically enough. For the North, the fact that Washington NY was not in immediate danger, combined with the distance between the capitals, would have meant less belief that winning the war meant merely marching a few miles south and batting away the Rebs while capturing Richmond. With more terrain between the two, the North may have settled down for a long war much earlier, and been much more prepared for the war as it actually occurred.
On the other hand, the Confederacy would now no longer have a huge barrier between its lands and incurring into the Union. The Potomac River acted as a major impedement to the eventual attacks into Fredericksburg and Gettysburg; with all crossings being under Confederate control, and no major natural features seperating Pennsylvania and Maryland, Confederate raids into Philadelphia or Pittsburgh may have been more common, and brought much grief to the Union.
Second is the matter of demonstrations and activism. New York, given its size, was given to major uprisings such as the Draft Riots of 1863, and various union/anti-union violence. Washington’s distance seperated it from these events; if those events had occured right outside of the Capitol… perhaps the government would have taken more progressive measures faced with the obvious violence, or perhaps it would have cracked down three times as hard on unions and anti-war activists. Hard to say, but I lean towards the latter.
Third is Civil Rights. The Civil Rights movement paradoxically benefited from the fact that Washington DC was a segregated city- photo-ops of “Whites Only” with the Capitol in the background, or the general embarassment of being an international capital that still had segregated housing, made it easier to convince average Americans that segregation was a national blight rather than simply a Southern one. Move the Capitol to a state without segregation laws, and it might have been harder to convince the general populace that the South’s Jim Crow laws in any way mattered to the rest of the nation.