If local population growth wasn't happening through prolonged periods of time, it means that something was preventing it. The idea that it was stopped by humans somehow successfully coordinating to limit their fertility and avoid the tragedy of the commons strikes me as implausible to the point of absurdity, for the reasons already mentioned.

I wholeheartedly agree.

Anything that makes people more phyloprogenitive will do the trick. In the long run, even behavioral mutations are conceivable, but cultural changes can also have a dramatic effect, and they act nearly instantaneously on evolutionary timescales. You yourself provide one possible answer: in the situation you describe, a mere cultural change that would shorten the breastfeeding period would, ceteris paribus, boost the fertility significantly.

Errr, yes, this happened. Approximately 10,000 years ago, in fact. Agriculture is a mere cultural change that shortened the breastfeeding period with a nearly instantaneous effect, by an evolutionary timescale.

The Malthusian principle says only that some resource constraint will stop further population growth, not what exactly that constraint will be.

Predation is not a resource constraint, yet it too halts population growth. The high rates of infant and maternal mortality that prevailed in all human societies prior to the past two centuries also limited population growth. Resource constraints, food or otherwise, are far from the sole determinant of population size.