Jerusalem is bound by the Status Quo, which predates the change, thus they never switched.Why is that? Why are they bound?
They do not want to start a war (it starts either way but still).
Well... In a word, no. Honestly, that's a rather sensationalist way of putting it, and not entirely fair.
The Status Quo binds all three major groups (Greeks, Catholics, Armenians), and the minor groups as well, to the same set of rules in governing what would otherwise be a totally chaotic situation in the Holy Sites. One could argue it's chaotic under the current agreement, but it more or less works. And it would be exceedingly difficult, under the current situation (or even as it existed in 1923) to fundamentally change the parameters of the agreement. Nor is there really any reason to. The Status Quo was carefully constructed to balance the needs of different groups against the physical spaces they commonly held. If it works, why fix it?
Also, it should be noted that the Patriarchate of Jerusalem has a bit more autonomy within the Armenian Church, as does the Patriarchate of Istanbul. It's their prerogative to do what's best for their local situation. And in this case, it means fundamentally molding their practices within the parameters of the Status Quo. The life of the Patriarchate is intrinsically tied to the Holy Sites. That's what makes it so special.