It's impressive how well they can find lack of evidence when they look hard for it.
The fact that there is a strong correlation is undeniable. Here are 3 links with charts that show the connection strongly.
https://www.kff.org/policy-watch/covid-19-deaths-among-older...
https://insights.som.yale.edu/insights/soaring-covid-rates-i...
And, one of the most striking to me, vaccination vs cases of counties in California:
https://www.nytimes.com/2021/09/15/briefing/gavin-newsom-rec...
That said, epidemiology is complex, and an attempt to reduce it to one factor is not valid.
ETA: Carl Bergstrom weighs on with additional insight into why this paper is bullshit: https://twitter.com/CT_Bergstrom/status/1444732537792708608