That much is true, but have you looked at a population map from that era? By far the vast majority of population, commerce, transportation and communication were in the EAST. That's where the nation was, that's where the capital would be. OK, so the OP is insisting on the capital not being DC, fine. I won't repeat my thoughts on THAT subject. The new capital might well be new and purpose-built like the old one, but it would still need to be somewhere in the east.
I'm not saying it will be in the middle of paradise. I think about it in three ways:
-Politically: the nation has just emerged from a devastating civil war between north and south, but the future of the nation is in the west. Keep people focused away from the sore spots and thinking about the future and maybe they become more productive. It means the South doesn't have to answer to a "Yankee City" and the Northerners don't get angry because they were not chosen over a potential rival. Build the whole city from the ground up and design it as you like, again along the Missouri River so it can also be a river port if you like.
-Defensively: The US had a paranoia of the British Navy and the UK in particular, visions of East Coast cities burning and Union Jack flags flying over various cities was vivid in the minds of civilian and military planner alike. Detroit is way too close to the Canadian border when the UK is not a staunch ally and most of the other East Coast cities are ports vulnerable to attack, at least DC has the Cheasepeake as a warning system. Missouri/Kansas/Iowa are the heartland and very safe from invasion by all but a truly continent-scale attack that would require just about the entire planet agreeing to take down the country.
-Monetarily: Custom-building new structures for the government will likely be cheaper than building in Manhattan, Phildelphia, or other established trading centers. If the capital were to be relocated they would need cheap land and lots of it, cheap labor was already plentiful and they could also attract development of the West by moving men and supplies out that way. Also it would bind the Pacific Coast closer to the rest of the country, the materials and resources of the West being the fuel for the engine that drives American expansion in the decades following the Civil War.