Move the US capital out of DC

I have thought for a long time that it might be a good idea to move the US capital to a more central location, which, two centuries ago, was the reason for building it where it is.

Two possible locations one could arrive at logically are the mean center of population, and the mean center of state capitals, which would reflect the distribution of Senators. Senators leaving a centrally located Senate chamber would leave in equal numbers to the four points of the compass. So, suppose you wanted to relocate the US Senate to a location where 25 of the state capitals were to the east, and 25 state capitals were to the west, 25 state capitals were to the north, and 25 state capitals were to the south. This would be an ideally central location for the 100 Senators to meet, the ideal location of a new Senate chamber. Where is this location? I couldn't find any document about this so I had to make it myself.

The median center of the state capitals is a rectangle bounded by
39.776N (latitude of Indianapolis, Indiana) on the south,
39.781N (Springfield Illinois) on the north,
89.644W (Springfield Illinois) on the east,
90.207W (longitude of Jackson, Mississippi) on the west.

In other words, the median center of the 50 state capitals is inside a rectangle with Springfield Illinois at the northeast corner and 39.776N 90.207W at the southwest corner. It is a surprisingly small area, due to a clustering of many state capitals at that latitude. The location is directly east of Springfield Illinois and north of St. Louis. The presence of a large city nearby provides airports, roads, rail, and other transport connections and services. This area is near the Mississippi River, which could replace the Potomac as the river running through the US capital, if it were to be moved over to the river. It is also not far from Hannibal Missouri, boyhood home of Mark Twain, and Quincy, Illinois. Specifically, it is located just north of Jacksonville, Illinois, off interstate 72, which runs west to Hannibal.

If we consider Washington, DC as a state capital, the median-ranked #26 on westernness is Springfield Illinois. Coincidentally, #26 on northernness is also Springfield Illinois! The US Senate would move into the city which is also the Illinois state capital, in the Land of Lincoln.
What about the US territories and commonwealths of Puerto Rico, Guam, the Virgin Islands, American Samoa, Northern Marianas and Federation of Micronesia? All of these could potentially become states and alter the center. All six of these are south of the other state capitals, and four of the six are west of the others. Together with Washington, DC, there would be 57 States, with the median being rank number 29 (28 states below, 28 states above). This would move the latitude to the line of 39.148N from Carson City, Nevada (if I am not mistaken). Adding four western states and two eastern states would bump the median state on westernness over one state, back to the Jackson Mississippi line of 90.207W. But somehow, the addition of these six additional states only serves to move the new capital south, two-thirds of the way from Jacksonville toward a location about 30 miles north of St. Louis.

The mean center of population for the United States is another candidate for a national capital. Washington DC was originally chosen as the capital because of its central and neutral location. The mean center of population went from east of DC in 1790, to north of DC in 1800, and has been moving farther west of DC since the 1810 census. The center of population spent 6 decades in Indiana before moving into Illinois for 3 decades from 1950. Since 1980 the mean center of population has been in Missouri. Locating a capital at the mean center of population after every census would mean leaving a trail of abandoned convention-like facilities across the midwest, unless tents like Mongolian yurts or circus tents were used.

I suggest we create a new national capital for the United States on the Mississippi River. Ideally, it could be located near St. Louis, perhaps within 50 miles, or 15 minutes by maglev train. It should be located on high ground, not flood plain. A large central maul like that of DC (or Red Square, Beijing) should be constructed as well. DC should either be made into a state or returned to Virginia and Maryland. The residents of the new federal district should retain the voting rights of the state which the land they reside on came from, to avoid what happened to the disenfranchised DC citizens. Government functions could be slowly moved out of DC over 30 years. A few functions could stay there. The DC area would become a national park with historical, educational, and other new functions.

The central location would make it easier for representatives to travel there, and easier for constituents to travel there to petition their representatives. It might put the government in closer touch with the problems at the center of the nation, such as the situation in New Orleans, or the crime problem in St Louis, which now exceeds that of DC, NTC, or Miami. At least for a while, the old lobbying networks of K street and the Pentagon system might be disrupted.

A new federal district could be carved from Missouri, Illinois, or both. Another alternative is to move downstream until the opportunity to make a capital of a Tri-State Area emerges. The nearest one is Illinois, Missouri, and Kentucky (near Cairo). A bit further downstream is the meeting of Kentucky, Tennessee, and Missouri (near New Madrid). Still further is Arkansas, Missouri, and Tennessee (Blytheville). The disadvantage is that these are really remote locations, but are getting closer to Memphis and the future mean population center, which is likely to cross into Arkansas. The remoteness could be solved by a high-speed maglev bullet train line that would link Chicago to St Louis, the new capital, and Memphis, the present route of interstate 55.