No US states started with such a provision. These came about in a wave after the defaults in the 1830-1840s when most US states had borrowed internationally (e.g. from London or the Dutch) and the US federal government refused to bail them out:
The first wave of adoptions among 19 states during 1842–57 is closely linked to the financial panic
of 1837 and subsequent economic depression. According to Wallis (2005) and Wallis and Weingast
(2008), the emergence of balanced budget rules should be understood as the demand of voters for more
transparent and realistic financing rules. All of the states that defaulted in the 1840s except Florida,
Mississippi, and Arkansas inscribed some kind of deficit restriction in their constitution immediately
afterward. [...] New states admitted to the union after the Civil
War generally included debt limits in their constitutions (Ratchford 1941, 122, whose explanation is
consistent with Wallis and Weingast’s).
Another wave of US state (but mostly local) defaults happened in the 1930s. Although the federal government didn't assume their debts... it did something different this time in that it substituted the falling state & local investments, so that by end of the Depression it had assumed the dominant share (and with that increased influence):
The 1930s saw another wave of defaults by local governments and the last default to be recorded by a
state. [...] There was a complete reversal in the relative shares of total government spending of
the three levels over the course of the Depression. Whereas in 1932 local governments spent 50 percent,
states 20 percent and the federal government 30 percent of the total, by 1940 local governments spent
30 percent, states 24 percent, and the federal government 46 percent (Bordo, Markiewicz, and Jonung
2011, 9, citing Wallis 1984). The period thus marks the ascendance of the federal government relative
to the states and, notwithstanding President Franklin D. Roosevelt’s instinctive fiscal conservatism the introduction of countercyclical demand management at the federal level. [...] The last state default occurred in 1933 when
Arkansas suspended payments on its highway bonds.
In general Keynesian economists say that US state's budget policies magnify the problem that the federal government has to "fix" during a recession:
Because state and local budgets are about 40 percent of total government spending in the United
States, fiscal policy is effectively shared by the levels within the federal system. [...] The size of this
effect can offset a substantial portion of the countercyclical movement of the federal budget position.
Krugman (2008), for example, refers to the states as the “fifty little Herbert Hoovers,” pursuing fiscal
contraction when Keynesian measures were in order as the United States was sliding into the “Great
Recession.” [...] in the aggregate state and local budgets do not help to stabilize the
macroeconomy during recessions; that role is played by the federal government in the United States.
Fiscal transfers from the federal government directly into state budgets, to help them fulfill federal
mandates and otherwise alleviate budget pressure, ameliorates the procyclical influence of the states during
downturns. The American Recovery and Reinvestment Act (ARRA) of 2009, for example, provided large
amounts of support to the states. According to the Bureau of Economic Analysis (BEA), the level of total
grants-in-aid to state and local governments in 2009 was $482 billion, $70 billion of which came from the
stimulus package. Federal support then rose to $532 billion in 2010, of which $100 billion was accounted
for by ARRA. A large part of the support was directed through Medicaid to cover the shortfall of revenues
at the state level. The rest was either spent in the education sector or earmarked for various investment
projects (see table 2). In 2010, according to the Congressional Budget Office (CBO), 75 percent of the
grants to states contained in the stimulus package were used to finance state deficits rather than fund
So while not assuming their old debt, the federal government took on new debt on behalf of the states' deficit during the Great Recession. Which kinda makes the states' balanced budget a bit of a shell game in such circumstances. The ECB notes however that US states' own rainy day funds (about 0.2% of GDP) were somewhat adequate during the smaller recession of 2001, but were quickly depleted in that event.
Also, the balancing requirement is fairly narrow for US states. For most it covers state employees salaries, but not their pensions for instance. The latter have been used more or less as a "piggy bank" in some states, totally some $1T and:
the 20 states with the lowest-funded pension plans saw the financial position of their systems decline steadily from 76 percent funded in 2007 to 56 percent in 2017.