Many of the issues you encountered are indeed not TK-specific.
The security issue is totally out of TKs control. It's the airport, period. And it must be a fluke, as I've never had the same experience when flying business, on TK or any other airline.
And the lounge at E... Check the LY forum for discussion of the E lounge. It's currently only used by BA and LATAM. Long story... but no other airline uses it. Indeed, it's high time for a Star lounge at TLV, but it's not a TK issue.
Why did they succeed? I have some theories. First, they were the first foreign airline to introduce decent schedules to/from TLV. If I need to get to, say, Berlin or Oslo, before the age of the LCCs I could connect in maybe a dozen cities in Europe, but I had to choose between departing TLV at 5am and departing in the afternoon but arriving around midnight. With TK, connecting in IST, I could depart at 11am or 2pm and arrive early evening: much more humane. And the same holds for the returns to TLV. Second, TK had excellent catering, including in Y, which is a winner for many leisure Israeli passengers. Third, in spite of the political differences between the two governments, Israelis feel comfortable around Turks, it's that similar mediterranean attitude. Add that to the low fares to Antalya and the all-inclusive deals there, and you will understand why Antalya, and Turkey in General, are preferred vacation destinations for Israelis. Finally, TK regularly flies widebodies to TLV, and is one of the only airlines that can take you from TLV to several European cities (and, of course, to most long-distance destinations) in lie-flat seats. And indeed, their fares used to be very competitive (less so now, I must admit).
I suppose this should provide some explanation.
Shuly