We haven't even fully buried the Krag yet, the very first US 30 cal military cartridge and the '06 is as healthy as it has ever been so the chances of the latest US military 30 cal dying in the next hundred years or so are about ZERO.
Some trivia about the 7.62 Nato/7.62 X 51mm/308 Win.
It was designed in response to the military wanting a shorter actioned and lighter battle rifle.
Bulk and weight of amunition was also a primary concern for machine guns.
It was designed for the use of all NATO countries many of whom wanted a shorter lighter cartridge than the '06 and the actions in which it came. Many wanted 7mm or 8mm or even 6.5mm but the US military machine bullied them into the 30 cal.
The 7.62 NATO as designed would and was intended to headspace and fire in all military 30-06 rifles including bolts, semi autos (Garand) and full autos, a criteria the military insisted on so their supply of weapons would not be rendered immediately obsolete.
As with all military cartridges, hunting of big game animals is not their first or last design criteria, but most end up being adequate, if not ideal, for the task of hunting and the 308 is no exception.
Is it dead or dying?
If you look at caliber offerings world wide, from all the gun manufactures I think you will find it ranks number 1 or 2 if my statistics and memory serve me correctly