Did the U.S economy flourish after national debt was eliminated by President Andrew Jackson in 1835?

Individuals are not nations; the analogy is specious. National debt has as much to do with individual debt as the national policy on capital punishment has to do with the individual policy on capital punishment, or the national central bank has to do with the individual bank.

Nations must manage a monetary policy. Nations have the option of creating or destroying money; it is criminal for individuals to create or destroy money. The flaw in your analogy is most strongly illustrated by the fact that there is no reason why an individual would want to destroy money, but nations regularly destroy money to avoid inflation. Inflation simply isn't a problem for individuals.

Individuals make choices about wealth - do I want to work harder to earn more money or do I want less money and more relaxation? Nations don't have that option - there is no employer from whom they can seek additional work. Individuals work and receive money in compensation for the value they create; if they are not paid for their effort, they can switch jobs or take the employer to court. Nations do not add value; their income is extracted by force. If they do not receive compensation, they cannot switch populations; they do not need to take anyone to court because they are the court.

When an individual's financial choices turn out to be catastrophically wrong, there are options - charity, or ultimately bankruptcy. The consequences of those options are borne by the individual. When nations financial choices turn out to be catastrophically wrong, there are no options. THe resulting suffering is borne by the population.

Even more serious, the world's economy is tied together in ways that individual finances will never be. If you run into financial problems, you might need to borrow money. If a nation runs into financial problems, they are more likely to employ monetary policies that risk the welfare of all trading partners (inflation, deflation, etc.)

The question reads like an excerpt from the debates between the Jeffersonian and Hamiltonian factions of the first administration. In that debate the Jeffersonian faction spent money hand over foot and continually blamed bankers for the fact that they were in debt. The Hamiltonian faction recognized that banking and debt was tool that permitted a country to flourish despite the impediments of chance and reality. Debt helps to cope with a bad harvest or an accident. Debt enables a nation to invest in the future. Debt enables a nation to borrow from a future strong economy to strengthen a current weak economy.

The analogy has a more serious flaw; it selects one policy from the complex set of historical circumstances and the panoply of policies implemented and attempts to use it for prediction. In 1996 the United states eliminated the US Tea Tasting Board; what were the economic consequences of eliminating executive branch functions?

Nations do not have the power to say, "I will gladly pay you Tuesday for a hamburger today...." - If the crops fail and the population is starving, it is sane and rational to borrow money to purchase food. If the country is attacked, it is sane and rational to borrow money to pay soldiers, buy boots and bullets and butter to defend the nation.

The primary advantage of having no debt is that it enables you to borrow money more cheaply. High debt can be constraining - it raises the cost of borrowing money. Zero debt is just as unwise as crippling debt. Debt is a tool that the nation must manage. Demanding zero debt is a simple, ideological policy.

I suspect that there is a national law "The simplest policy is not always the best; it is more likely to be the worst than the best."

Per your other question on the topic, the US was hardly shackled by public debt at the time. It was a few percent of the GDP, i.e. on the order of a rounding error compared to other major drivers of economic activity at the time.

There was prosperity in 1834-1836 followed by a financial crisis in 1837, which economists attribute to external factors beyond US control much more than to Jackson's Bank War.

As to whether having zero debt is good or bad, there's precious little no public debt experience to draw on, be it in the US or elsewhere in modern times, so you can't really compare how not having debt fares against having some.