Frequency response of a rolling linear regression
This is a statistics question and should not be asked here. But I will try to answer the question as best as I can (it won't be the best answer). First off, I don't see what the phenomenon you are observing has to do with frequency response or LPFs. Low pass filters makes data smoother, which is not what you have. Instead you have a situation, where estimate for one set of data differs greatly from a other set of data. The OLS estimator also is not an MA process, the formula being cov(x,y)/var(x). The filter described in the paper is not a linear time invariant filter.

Indeed, if you regress a line on white noise process or linearly growing process, the regression line should not change slope significantly as more data is gathered. Instead as the amount of data grows you should be more certain of that the coefficient is right, generating greater variance for the rolling process, but on average the slope should be right.

What you have here is not a process that satisfies Gauss Markov assumptions. As can easily be seen the expected value of the error is not zero, and there is heavy auto correlation. This means that the error term is correlated with the independent variable, producing a biased estimate. Besides that the data is not randomly selected.

Most importantly, the data is increasing as a function of time. This generates a so called spurious correlation. Since Y increases faster than X as time passes, the beta becomes upwardly biased. You can model autocorrelation with an MA process, but this is not really important...

I am sure statistics stackexchange (crossvalidated) would provide you with much more mathematically precise answer and a better explanation of what the phenomenon. One thing fix that might help is to difference both of the variables, which changes the autocorrelation structure of the error. This can be equally viewed as a (1-1) MA process applied to both of the variables, which is a comb filter at a high frequency, producing a high pass filter below the nyquist frequency. But the data has such a non-standard structure that I wouldn't be so sure that the fix will help you.