I thought it may be helpful to see the derivation of the discussion, and why it's relevant. Anti-vax groups claimed that the Vaccine would alter your DNA, some even went as far as claiming it was a plot to strip us of our human rights, because we would no longer be genetically human.
I can't seem to find a direct quote of any scientist saying that it is impossible, though a bullet-point on the CDC site about vaccines does use the word 'cannot', but bullet point summaries aren't exactly famous for explaining nuanced concepts in detail, are they? I would certainly be open to reading an entire direct quote in context if anyone could find it. I think it's highly likely that in order to address the wave of vaccine 'DNA' misinformation, rather than provide a soft response like 'There is no evidence that mRNA vaccines alter DNA', or 'It's plausible that mRNA vaccines can change your DNA, but not probable', some poor science communicator somewhere (Again, I can't actually find a direct quote of a credible expert saying that it is outright impossible) opted for an assertive response and just said it was impossible.
Regardless of whether we can attribute the 'impossible' claim to anyone relevant, this seems to be a semantic response to the fact-checking of anti-vaccine misinformation.
'The COVID Vaccines alter your DNA, you won't even be human anymore!'
'Sorry, that's impossible'
'Here's a scientific paper that says it's possible'
'Well, yeah. It's not very probable though, there are a few other more plausible explanations'
'So you lied, GOTCHA! NUREMBERG 2.0!' (I wish I was joking, if you scroll to the post below the good-doctor's and click the first citation, https://www.flemingmethod.com/ Sidebar: The hot-Fleming method also sounds like a euphemism for having someone poo on your chest)
This is my impression of the broader context of the discussion, but hey, I'm not an expert. I snuck a peak at the paper and I'm sure I don't know enough about the nuances of genetics to really draw a conclusion from the data contrary to that of experts in the field.