I agree with the approach taken here. The most dangerous conjecture so far was taking one single baffling iteration of an experiment as possible (fine if we are to construct a road map) and dumping a whole bunch of extraordinary results on top of it. We jumped straight to photons have mass on the first page!
Anyway I think this would have to be pretty close to the starting point. The whole implication can't be to throw out c is the speed limit but that observed photons don't travel at c. This may lead to that we may have to redefine how we interpret 'vacuum'. This, I think, would imply that neutrinos have mass (i.e. not affected by the whole vacuum issue as much, like neutrons scattering light in a nuclear reactor due to them moving faster than the photons in a non-vacuum)- something we are far more prepared for than 'c doesn't hold, let's scrap sr/gr'. In any event, it would be a very, very long and messy path of retrofitting theories before we can even consider scrapping any part of sr/gr. We have to address the 'frame' the neutrino travels in. Do we know enough about the neutrino to claim that it actually moved ftl. It may have 'appeared' to move ftl but we know that ftl travel is possible just not locally under gr.
If (a remote chance) this is true I'd bet it is far more likely going to have implications on the nature of the neutrino, possibly even the graviton (another very long shot), than forcing a rework of a century's worth of work. So if you are keeping score at home we are at (long shot)^4, and we haven't even dealt with (long shot) so lets not get our panties in a bunch here.