Placeholder while I work out a decent response to this, but for immediate interest OP you can get a good portion of Hamilton's reasoning in his own words from Federalist Paper 30 for the core need to have a centralized collection of revenue and credit.

link here: http://avalon.law.yale.edu/18th_century/fed30.asp

additionally you can find a semi-contemporary interpretation of Hamilton's debt theories and how they've been warped or continued in modern times in John Gordon's *Hamilton's Blessing: The Extraordinary Life and Times of Our National Debt *

Hamilton's ideas on a large national debt sustaining economic growth were a result first and foremost of his observations in the practice of the nation's British forebears. British Debt had its founding in the aftermath of the Nine Years War, a coalition against absolutist france at the tail end of the 17th century (not to be confused with the War of Spanish Succession, which would follow it promptly near a decade later and involving many of the same principle partners).

Prior to 1694 British war expenditures were financed near exclusively through direct taxation levies, but the raw cost of the conflict led to the need for a new institution. The first prosaic attempt at a central bank was thus induced, with the institution being granted a great many fiduciary privileges in return for loaning the British government ~1.2 million pounds at an interest rate of 8%. In a minor digression, shortly after the foundation of this bank and in the wake of the War of Spanish Succession, the South Sea Company would be formed and would function to service the debt in place of the Bank of England - however the series of scams, failed negotiations, and idiotic trading ventures carried out by the Company are a debacle meriting its own post or multiple such replies.

To get back on topic, the ability to finance current expenditures beyond the immediate means of direct revenue, balanced against growth to offset the cost of interest on such loans, allowed the British government to (in the eyes of people like Hamilton) spend their mighty rival france into submission as the 18th century progressed. The irony of this observation when a major cause of revolt for the americans was revenue raising measures taken in the aftermath of the Seven Years War (also known as the French-and-Indian War) should be noted and chuckled at healthily.

Banking in Britain also allowed for the ready growth of Fiat Currency, demand for which spiked in the initial phases of british industrialization circa 1750ish. Fiat Currency helps with liquidity and ease of transfer of wealth, which in the accelerating economy of an industrializing nation is a major chokepoint for growth. A core element of Britain's supplier of Bank Notes and liquid currency was the Bank of England, who's roots and prime function was still servicing the national debt.

Great Britain was also able to use this expansion of available funds to help finance its initial forays into infrastructure projects in Scotland through the 18th Century, though more drastic expansion of infrastructure programs would be delayed until after the American Revolution.

So in observing what was still clearly one of the leading powers of the world, the former suzerain overlord of the United States, and the closest economy and cultural tie the States had, Hamilton would see a nation that had used a National Debt to help survive the rapidly growing costs of a modernizing army and the demands of higher cost warfare, which had used these tools to help stimulate economic development beyond that of its neighbors, and had done so while remaining reasonably solvent over its recent history.