Strengths: This is an original time series analysis pertinent to questions of current national interest.
Points for consideration:
Perhaps averaging will make the delay signal harder to find. Also, cross-correlations before detrending may make the lag harder to see because of the trend.
In the AIC table, ARIMA(5,1,4) is not the lowest AIC.
Log-transforming data might help (or, possibly, square root).
You could compare different lag relationship models by AIC (or likelihood ratio tests) in a regression with ARMA errors.