The chances to interact is proportional to the cross section, which for an electron is around 0.6 barn. In comparison, the cross section for photon-photon interaction was measured as 70 nanobarn, so there's a factor of 10^8 difference in probability of interaction. Photons doesn't outnumber electrons by such factors.
Further, there are plenty of atoms in the outer layers of a star, and these have something like another 10^8 advantage over electrons.
One day I was watching this guy on YouTube whose job is walking around selling peanuts at a baseball stadium in Chicago, blue collar local accent in full tow. He starts to chatter about in a casual, simple, mundane way about how his day was going.
Except, it wasn't a baseball peanut barker, it was a Nobel prize winning particle physicist, it was Feynman. If he would have sold peanuts part time at a baseball stadium for five years maybe no one would have noticed anything out of place.
It's not a criticism, it only raised his rank on my list of admired scientists. I found the humility, his passion for making hard things as easy to understand as possible, his ability to just be a regular guy around people, really a beautiful thing.
(relevance: he's cited in the paper)
Now let’s consider what is going on. We have two photons in and two photons out in different energies and directions although the total energy and momentum must be conserved. Anything could be happening in between, so let’s stick with known phenomena.
We know about photon -> particle/antiparticle, and the reverse. We also know about particle/antiparticle -> dual photon, and the reverse, which must exist by time symmetry. So it’s not hard to see that dual photon -> particle/antiparticle -> dual photon interactions are very possible.
So how do we get scattering? First, if the input photons are opposite momentum, then the output photons may be in any opposite directions. Actually, any two incident photon momenta define a cone of output direction possibilities, and the lifetime of the +/- particle system translates that cone along the total momentum vector. Thus, many observations are possible at the detector.
However, I don't see what clarity you get from bringing in quantum statistics to the discussion, the blobs at the centre of the diagrams represent both fermionic and bosonic particle exchanges: https://arxiv.org/pdf/1601.07001.pdf