While these rifles may be marketed and sold as 308, but they were indeed chambered for 7.62X51 NATO, and its lower pressures. They were intended as reserve weapons and not as first-line weapons, and as such the concern for their strength and viability in high-volume shooting was not a consideration.
BTW, few if any of these rifles are 100+ years old as is often heard, most of the older rifles were scrapped after the Spanish Civil War, so the dates typically seen on these rifles is from the 1930s on.
I have had several 1916s over the years, 7X57 and never 7.62X51, and fired 7X57 military ammo in them with no negative effects. I have also heard of some that were so soft that they would be unsafe with even 7X57 ammo. If you feel a need for a M1916 as a cheap shooter, definitely be nice to it.