The short answer is because most senior Israeli politicians in the 1950s through early ’70s (such as David Ben-Gurion, Menahem Begin, Shimon Peres, etc.) were foreign-born and couldn’t do the guttural rolling Rs, so used trilling Rs instead, as in their native Russian or Polish—whereas modern-day leaders are almost invariably native Israelis. (The exceptions were Golda Meir and Abba Eban, whose mother tongue was English, so they spoke with painfully Anglo-Saxon Rs.)
But as we say in Hebrew, בואו נעשה סדר בדברים—let’s clear up some things, as there’s a common confusion here, even among Israelis themselves.
There are two types of rolling Rs: the alveolar trill (let’s call it the ATR, for short) made by the tongue at the front of the mouth, as in Spanish, Italian, Greek, and Arabic:
and the voiced radical pharyngeal trill (call it the VRPT) which is made at the back of the throat:
This is much rarer, and in fact, almost uniquely Israeli.* The VRPT is said to have originated in Yiddish speakers of certain parts of eastern Europe, and inherited by their native-born children. But the VRPT is not the same as the French R—it’s a more like the Rs of Édith Piaf in her immortal Je Ne Regrette Rien—which, though magnificent, is not characteristic of the vast majority of French speakers:
Even among Israelis, the truly rolling VRPT is rare—which is why, when they talk about ‘rolling Rs’ (רי”ש מתגלגלת–’resh mitgalgelet‘), they’re invariably referring to the ATR, not the VRPT. For starters, it’s characteristic mainly, if not exclusively, of people raised in the kibbutzim and moshavim (rural communities)—especially those in the Galilee, for some reason (presumably those founded by PTR Yiddish speakers). Listen to speeches by Moshe Dayan, Yigal Alon, or Yitzhak Rabin, and you’ll see what I mean.
Today’s politicians, being virtually almost all native Israelis, are capable of VRPTs to some extent or another. But in practice, most native Israelis use a shortened version of the VRPT (the plain pharyngeal trill R, or PTR): it’s still in the throat, but it’s a single r, as it were, as opposed to a rrr (the pharyngeal equivalent of the Spanish caro vs. carro). Often, however—or all the time, with some Israelis—it comes out like the French R.
Until the 1970s or so, Israeli broadcasters, singers, and theatre performers were required to use the ATR, as it was thought to be clearer, and quicker (which is critical in singing). However, in the past twenty years, and certainly in the past decade, there has been a grassroots ‘revolt’, especially among younger broadcasters, against the ATR, and a tendency to use the more natural PTR in broadcasting, theatre, and singing. This is neatly explained in the following YouTube video: it’s in Hebrew, but even if you don’t understand what people are saying, you can hear the difference between the old, ATR, pronunciation, and the modern, PTR one:
* Arabic has something similar in its letter *ghayin* غ, but that is a combination of R, hard G, and guttural ayin—which is why the name *Khaddafi* is sometimes written *Ghaddafi*, and *Gaza* in Hebrew is *Azza*