It seems like population historically can overshoot productivity.

I think this almost certainly is not the issue. In modern times there's an inverse relationship, including in developed countries, between fertility and income. So the people most economically capable of having children are the ones choosing not to.

I don't know if the same was true in Rome, but indirect evidence would suggest it may well have been. Here [1] is a selection of some of the laws that were created to motivate fertility. They definitely target what were probably higher class citizens - concubinage was legally recognized, inheritance was one of the main targets of penalties, ranking of politicians was determined by their number of children, and so on. There was even apparently some angle shoot where people tried to marry very underage girls to avoid the penalties against unmarried men while also having a legal justification for having no children 'yet'.

[1] - https://www.csun.edu/~hcfll004/AugMarriage.html