I'd do a back-of-the-envelope calculation using BA's Executive Club vs. Cathay Pacific's. CX Silver is Oneworld Ruby, which won't get you lounge access except to CX lounges before CX flights, which you'd get anyway if traveling in business class.

From the sound of your travel patterns it sounds like you'd have little difficulty in reaching BA Silver (Oneworld Sapphire) which would allow lounge access across the network, even to AA Admirals Clubs on domestic itineraries in the US, something that even AA elites don't get. You'd have to fly at least 4 segments on BA metal (any class,) but that wouldn't be hard flying from TLV.

Depending on how often you fly long haul, you might find it quite easy to make BA Gold/Oneworld Emerald, which would bump your perks even more.

Generally award availability on BA is pretty decent, although of course you'll confront the dreaded BA fuel surcharges on award tickets. (They don't call it that anymore after getting sued in the US over it, but a rose is a rose.) But some Star Alliance carriers do the same thing.

But the analysis wouldn't take too long, and I'd bet you could easily find BA a better fit than CX for the long term.