Activity analysis was a thriving research area in the 1940's and 1950's. It was fruitful enough to earn the Nobel prize for Tjalling Koopmans. But it seems to have been abandoned altogether. Mas Colell, Whiston and Greene (1995), Microeconomic Theory, Oxford University Press, only devotes a seven page appendix to it and refer the interested reader to Gale (1960): The Theory of Linear Economic Models, McGraw-Hill. Based on this, my guess is that not much additional work has been done in the area since 1960 and I wonder why.
In Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society: Series A (General), 120(3), 253-281. which is widely seen as the starting point of systematic Efficiency and Productivity Analysis, the author writes, referring to what he is about to present:
Similarly, although the treatment of the efficient production function is largely inspired by activity analysis (see for example Koopmans 1951), no reference is made to this in the exposition. The professional economist can easily draw the necessary parallels for himself, as indeed, he can note the similarity of the measure of "technical efficiency" and Debreu's "coefficient of resource utilization"(Debreu,1951).
I would suggest that "activity analysis" has been subsumed into "Efficiency and Productivity Analysis", eventually living someplace inside Data Envelopment Analysis.
As for the reason why it did not stay in the "general microeconomics core" I believe it is because it was not a framework for economic models (that include behavioral hypotheses, incentives etc), but mostly a framework for quantitative measurement and analysis. As such it was more of a tool for applied decision making than for providing theoretical insights.