I think its very clear,
Due to the amount of parallel flights between EL-AL and Israir/Arkia (esepcially in the summer), If you dont fly EL-AL you are not entitled to use
the lounge, This is in order to prevent crowded lounge...

As why they dont mentioned other airlines, I think they dont want to be drafted into massive complains like they had few years ago when they did
such a restriction on all airlines.

They never check my ticket at the enterance to the lounge, I simply smile and continue to the Business/First, If someone (doesnt recognize me), I say i am TPL...

(was flying friday with BA and went to the LY lounge, No one said anything)