The most concerning scenario would entail the government failing to make scheduled payments of interest and principal on existing US debt because the government would have run out of cash to make those payments.
A default in such payments would damage the standing of the federal government in financial markets and diminish the perceived trustworthiness of US debt in the eyes of worldwide participants in financial markets. Global financial markets might plunge into uncertainty and chaos, with unknown—and unknowable—consequences.
The damage would likely linger for a long time, raising financing costs for the federal government in the future, potentially crowding out available budget funds for the military, Social Security and Medicare, and running all federal agencies.
Not raising the debt limit by the X date is not like a government shutdown, which occurs when annual funding for ongoing federal government operations expires and Congress does not renew it in time. Yes, government shutdowns have been costly to American taxpayers and caused disruptions, with the longest one lasting 35 days in 2018-19, but the federal government has not experienced a major default on its debt in recent history.