ARIMA: Understanding how time series analysis is focused on mathematical properties as opposed to best forecasts Rob Hyndman states:
"The paper describing the competition [M] (Makridakis et al, 1982) had a profound effect on forecasting research. It caused researchers to: ... treat forecasting as a different problem from time series analysis"
and
"The paper describing the competition (Makridakis et al. 1982) had a profound effect on forecasting research. It caused researchers to: focus attention on what models produced good forecasts, rather than on the mathematical properties of those models; ...; treat forecasting as a different problem from time series analysis. These now seem like common-sense to forecasters, but they were revolutionary ideas in 1982. Even today, I often have to explain to other academics why forecasting is not just an application of time series analysis.".

The author indirectly states that time series analysis is focused on the mathematical properties of models rather than on models that produce the best forecasts. Restricting our focus to univariate time series analysis with ARIMA, what does it mean for ARIMA time series analysis to be focused on the mathematical properties of models (as opposed to focused on best forecasts)?
Hyndman, R. J. (2020). A brief history of forecasting competitions. International Journal of Forecasting, 36(1), 7–14.

I don't think there is (or better: was, at the time of the first forecasting competitions) a true opposition between getting good forecasts and focusing on mathematical properties.
Rather, my reading of the literature would be as follows: statisticians and mathematicians looked at time series and very understandably focused on the data generating process. It's probably a fair assessment to say that this is a natural first reaction for statisticians. They quickly thought about AR and MA processes, combined these, added integration and ended up with ARIMA processes.
These were, on the one hand, fertile grounds for mathematical analysis. You could prove all kinds of theorems about stationarity, unit roots, estimation, identification of correct ARIMA model orders and so on. And on the other hand, if your original DGP was ARIMA, then the correct ARIMA model would give you MSE-optimal forecasts. This in turn increased interest in identifying that true ARIMA model.
More precisely, as Richard Hardy points out this would hold under perfect estimation precision, i.e. never in reality. Under more realistic imperfect estimation precision, an ARIMA model with lag orders different from these of the true DGP may generate more precise forecasts due to the bias-variance trade-off. Yes, statisticians were aware of this - but this again offers interesting avenues for research, in terms of asymptotics, and of quantifying the discrepancy between the MSE of true and misspecified models.
Extensions to (G)ARCH, VAR, VECM etc. followed, and there is again indeed a lot of ground to cover here, witness any time series analysis textbook.
There was no point where people decided to focus on mathematical elegance over good forecasts. Rather, it was a point of having found a framework in which both mathematical elegance led to good forecasts.
The problem was (again, let me emphasize that this is only my impression) that people forgot about that crucial assumption that the original time series was generated by an ARIMA process, or that it could at least be well enough approximated by one. If you want to be uncharitable, it's a case of having a hammer and all (forecasting) problems looking like nails, or of searching for your lost keys under an ARIMA lamp post, because the light is best there. This, it seems to me, was the intellectual background against which we have to see the early forecasting competitions in the 1980s, which for the first time focused on forecasting performance in a model- and DGP-agnostic way. The hammer-wielders saw people with screwdrivers and did not understand how they were going to get their nails into the wall... until they saw that there were screws among the nails they had been picking out of the toolbox.
Let me emphasize once more that this is only my understanding, not based on any particular research.
