Technically, these rifles were rebarrelled to 7.62 NATO (7.62x51), not .308 Winchester, which is not quite the same.
All the Israeli 7.62 NATO rifles I have seen were converted K98k's that Israel got in Europe in the first few years of the nation. They were, of course, in 8mm (7.9) and Israel made ammunition for them. Later, when they adopted the FAL in 7.62 NATO, they rebarrelled the bolt actions to that caliber. (They actually issued the Mausers during the six-day war when the FALs failed in the sand.)
Other nations had Model 98 Mausers converted at about the same time, and many of those are now on the surplus market.
I have found that Spanish receivers tend to be soft and would prefer the Israeli rifles to the Spanish, even the Spanish Model 1936 (Mauser 98) rifles. The Model 1916 is simply an upgraded Model 1893.
The old type Mausers (1893-1896 Models) are very unlikely to "blow up", but the working pressure of the 7.62 NATO exceeds the design pressure of those rifles; the working pressure of the .308 Winchester is even higher. The guns will not "blow up" but some of the receivers (especially the Spanish) will batter or stretch under the higher pressures. In effect, every shot is a "proof load" for the old rifles.
Jim