Yes, that's the way accelerators detect neutrinos and also how they'll look for supersymmetry. However in systems involving only massive charged particles and photons the photons can carry plenty of momentum and are experimentally observed to do so. But the only way the momentum of one electron can be transferred into another (for energies below 90GeV) is by it emitting a photon of that amount of momentum. For instance suppose you have two electrons heading at one another, one with momentum p and the other q. They scatter and come out with momentum p' and q' and by conservation of momentum you'd have q' = q-(p'-p). The main contribution to this process is the tree level exchange of a photon and it'll have momentum p'-p. It carries the momentum they exchange. When you go further and do loop contributions you encounter photons of momentum as high as you wish, its part of the reason you have to renormalise. So even in electron-electron scattering theory and experiment disagree with you. And then there's hard Compton scattering, an Xray photon slams into a charged particle and gives it a huge amount of momentum. Its like the photo-electric effect but more pronounced. And this means they carry negligible momentum because.....? Their momentum/energy ratio is independent of how much energy or momentum they can carry. So why are you claiming you've demonstrated them to be wrong, which is the implication of momentum conservation violation you're talking about. How can you reach conclusions like "Its simply IMPOSSIBLE for momentum to be conserved if photons are the only momentum carriers that are created!! It is also impossible for the hypothetical momentum carriers to be massless. ". You admit you're cutting corners and not using the full models and yet you're proclaiming those models are either wrong or don't do what people think they do, based entirely on your very superficial back of an envelope guessing. I disagree with your application of them to this problem and I disagree with your conclusions as they are based on simplifications and mashing together different models, some of which aren't terribly appropriate. When experiment disagrees with you you're wrong. QED models bremsstrahlung processes with enormous accuracy, its tested every second of every day there's an accelerator running somewhere and QED says momentum is conserved, which is what we see. We've seen in experiments photons with momentum of the same order of magnitude as massive particles. You are arguing that not only is QED wrong but the universe is too.