DC didn't exist in 1787. They did foresee a federal district, however, in Art. 1.
The general argument is that the national government must be responsible for its own maintenance and safety, rather than relying on the loyalty and good character of a state (not an objection to be taken lightly back then).
It's also a prestige thing - everybody wanted it, nobody wanted anyone else to have it.
Reminds me of Douglas in 1860, after he figured out that he had lost and that the South was going to secede. He toured the soon-to-be Confederacy trying to reverse the events that he had helped set in motion.
That's debatable. When you go down the turnout road, you tend to stop at a convenient road instead of folowing it to its end, which is that any turnout short of 100% is, by definition, unrepresentative of the will of the people.