That's baloney. The old ML adage "there's no data like more data" is as old as mankind itself.
Most methods research went into ways of building beliefs about a domain into models as biases, so that they could be more accurate in practice with less data. (This describes a lot of PGM work). This was partly because there was still a tug of war between CS and traditional statistics communities on ML, and the latter were trained to be obsessive about model specification.
One result was that the models that were practical for production inference were often trained to the point of diminishing returns on their specific tasks. Engineers deploying ML weren't wishing for more training instances, but better data at inference time. Models that could perform more general tasks -- like differentiating 90k object classes rather than just a few -- were barely even on most people's radar.
Perhaps folks at Google or FB at the time have a different perspective. One of the reasons I went ABD in my program was that it felt industry had access to richer data streams than academia. Fei Fei Li's insistence on building an academic computer science career around giant data sets really was ingenius, and even subversive.
I definitely remember that bias about neural nets, to the point of my first grad ML class having us recreate proofs that you should never need more than two hidden layers (one can pick up the thread at [1]). Of all the ideas clunking around in the AI toolbox at the time, I don't really have background on why people felt the need to kill NN with fire.
[1] https://en.wikipedia.org/wiki/Universal_approximation_theore...