I'm not a biologist, so I could be completely wrong here, but I disagree with you.
First of all, plasmid != viral vector. That's why the Nature paper you cited had to use electroporation to introduce the payload. As J&J is a viral vector vaccine, I would not be confident in using this paper to make the argument that an adenovirus-based vaccine could modify a host's DNA.
Second, your article does not make the claim you think it does. Specifically, the discussion says:
> Using simple intramuscular injection, the vast majority of plasmid DNA that persists is extrachromosomal, and the frequency of integration, if it occurs at all, is negligible.
> However, even if the residual plasmid in the gel-purified genomic DNA did represent integrated plasmid, one copy... would be at least three orders of magnitude below the frequency of spontaneous gene-inactivating mutations...
That is, the delivered gene does not integrate directly into the genome itself, but rather stays in the cell. To detect whether the gene stays in the host cell, the article compares molecular weight and uses a PCR test. Both methods do not tell us whether the gene inserts itself into the host DNA; one way to detect that would be to sequence the subsequent DNA, which would be monstrously expensive (because the modification incidence is extremely low).
The only way that a gene might integrate itself into the chromosome is if during DNA transcription, an error occurs and the foreign DNA is merged onto the host DNA. But this would be incredibly rare because it would require most base pairs to match up between the host & foreign DNA; otherwise, the cell would attempt to repair itself or induce apoptosis.
And even if the resulting base pairs match up, the resulting DNA might not have any behavioral differences because it's (approximately) the same sequence of characters. (And you would need another transcription error later on that happens to reduce the non-modified strand by approximately the same length. That's also extremely unlikely.)
Fourth, the Pfizer and Moderna vaccines use a mRNA-based vector, which (in my limited knowledge) is very difficult to integrate into the genome. The most plausible mechanism I could imagine would be:
1. (Optional) an enzyme which converts a modified nucleoside mRNA strand into one that mimics the mRNA chemical structure that is compatible with a reverse transcription enzyme [1].
2. A reverse transcriptase to convert the mRNA strand into a (foreign) DNA strand.
3. A restriction enzyme that cleaves the host DNA suitably so that the foreign DNA can be inserted.
4. A matching ligation enzyme which actually inserts the foreign DNA into the the host DNA.
All these above enzymes do not naturally occur in humans, so DNA modification through this mechanism would be incredibly rare.
[1] https://link.springer.com/protocol/10.1007/978-1-4939-6481-9...