GDP provides a rough measure for how fast a country could drain its debt—similar to looking at debt-to-income ratio when issuing mortgages to individuals.
Nations also have debt to income ratios. We should be comparing those instead of debt to GDP. If you've ever seen one of those UK is poor than Alabama memes, you know that GDP is a poor and tendentious measure of well-being because UK quality of life measures put it above all about the richest states in the US.
Income through taxes is roughly a percentage of GDP.
You could also just compare interest spending vs budget, and lots of people do. Spending on interest is roughly $1T out of a total $7T with income of $5.23T
$1T of an incoming $5.23T is pretty concerning. Especially given projections that the $1T is likely to go up significantly over the next decade.
If someone has a debt it's entirely sensible to compare that to their yearly income, of course this tells you something. GDP is a proxy for potential yearly income in taxes for the government.
GDP is also an amalgam of various indicators of general economic activity: Consumption + Investment + Government Spending + (Exports - Imports). It might be all in dollars, but it is kinda like adding $X of Apples with $Y of Oranges.
Its good as a rough score to do relative comparisons between countries (and actually Debt/GDP is useful in that sense too), but as an absolute amount it doesn't mean all that much.
What matters is how much the debt servicing costs versus government revenues. Also how much that debt is growing (deficit) and/or what it would cost to reduce it.
But there's not much of a consensus around what is too much or too little.
I suppose 100% Debt/GDP is a good arbitrary number to raise the alarm, but it doesn't mean much on its own.
A quarterly debt to revenue ratio would be a lot more illuminating. There are very good reasons not to care if that ratio goes out of whack for one or two quarters. But any bro divergence that persists is bad.
This ratio isn't just better because the units match, you don't have to adjust the units for growth and inflation.
Miles per gallon is another example. It's a useful number.