Photon pair production in electron-positron annihilation is necessary to conserve the linear momentum.
Conservation of momentum in two body scattering is simple to understand in the center of mass system. In the center of mass system the electron and positron have equal and opposite momentum. The resulting particles will have to be at least two, so that the generated momementum will add up to zero.
In such discussions the electron-positron pair collide with equal and opposite momentum, if they collide without the above condition then only a single photon can conserve the momentum (or two or more photons with different energies).
Wrong , a single momentum cannot conserve the momentum in the end products. There are continuous transformations from the center of mass, so at least two particles will have to appear in the lab too.
Another question is why more than two photons are not produced in the symmetric collision of electron-positron.
In this paper three and four photon annihilations are studied at DESY.
LEP experiments measured e+e- annihilations to an large number of particles, for example this three jet ALEPH event. It all depends on the available energy , and LEP had very high energy e+e-.