If there was no teachers union, everything would depend on individual school administrations.
So there may not be an obvious causal link, but what has become clear is that good safety protocols reduce the rate of spread, and that implementing good protocols requires ongoing effort. I have no doubt that, in this situation, being unionized is better than not.
One cause was presented: more PPE (presumably demanded by union).
The study, if you read it is mostly based on NY data. It specifically states as much in the limitations:
> this study may not be generalizable outside of the State of New York.
I personally think the study should have published the raw data as well as the aggregate data.
But the study seems to be quite open about it's limitations, listing one particularly limitation which to me is the greatest one if you want to critique the study:
> We are therefore unable to adjust for the possibility that unionized health care workers may transfer residents to hospitals earlier and thereby reduce their facility’s COVID-19 mortality rate.
That doesn't mean they are political.
The study is about New York. The comment is trying to make this a 'red state vs blue state' thing. There's no reasonable basis for such thinking if you actually read the study.
Aka, Union vs Non Union means different practices in aggregate and studying the results of those differences is valuable real world data.