The first thing that someone would need to do is clarify exactly what the Gedolim think about this matter. The fact that there are signs all over Yerushalayim with names on the bottom doesn’t mean anything. There are kana’aim out there who will twist the Gedolims’ words for their own agenda. It could be that someone asked the Gedolim if it’s preferable to volunteer for a shomer-Torah emergency organization rather than MDA, and they answered yes …
If it is indeed true, I imagine that there’s a back story here … I also have trouble understanding how it could be really assur to volunteer for MDA.