I'm not a statistician, but is it valid to take a correlation with a moving average? By definition, the moving average throws away some of the scatter, so the correlation will be likely to be larger. And you've already thrown away a lot of the scatter by taking yearly averages. 
 
FWIW, I generated a 0/1 random variable for each month in the range and carried out the same analysis as agnostic to obtain Kendall's tau for the raw data and 3 and 7 year moving averages. Correlations appeared to increase with the size of the averaging interval, and p values decreased (although most were above 0.1). Here are the scatterplots for the first two runs: 
 
Run 1. 
 
Run 2. 
 
By definition, there is no trend in these datasets, yet spurious patterns can clearly be seen in the 7-year averaged scatterplots, even more clearly than for the Playmate data.