No it wouldn’t. It would be bloated by the number of people with serious disabilities, health conditions and retirees. As your population ages you would get a bump in unemployment that tells you nothing about the demand for labour.
In my first Economics subject in high school they told us to never look at unemployment without looking at the participation rate with it.
I think it would be useful to know if a country has a much higher dependent population than another country. It could highlight dynamics that get ofuscated when some countries put large parts of their populations under the "unemployable" label or quietly pushed into black markets, making official unemployment numbers impossible to compare with other populations.
That's called employment-population ratio and it is tracked by the BLS. It's actually a bit more valuable to track prime age employment-population ratio to factor out students and retirees. But it's still a broad indicator.