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If I'm reading this right, they calculated the rate of supernova eruptions, and looked at how close they would have to be to Earth to trigger an extinction, and they came up with "two extinctions at that rate".

And then they looked at the history of extinctions, and said "probably those two".

So the headline seems like a bit of an overstatement of the degree of certainty...

Yes, completely overstated. The 2009 paper on the supernova causing the Ordovician extinction (https://arxiv.org/pdf/0809.0899) has always been viewed skeptically, even for a paleontology paper.

The Ordovician extinction has always been contentious. There's evidence of a severe ice age, but ocean water temperature proxies don't line up. There's even a "It didn't happen" faction among paleontologits. There's probably 4 other plausible theories for that particular mass extinction, including 2 kinds of volcanism, dust influx from the L-Chondritic parent body breakup causing the ice age, land plants drawing down atmospheric CO2 causing the ice age, a meteor impact in Australia, and to my mind, a vastly less plausible explanation, "true polar wander": https://www.nature.com/articles/s41467-022-35609-3

You forgot the very important “study suggests” part of the headline. Also, the study doesn’t even suggest that anyway, it just suggests supernovae close to earth could maybe be happening at a similar rate as mass extinction events.