There is potential for adenoviral vectors (e.g. J&J) to change DNA as well via multiple mechanisms. Native mRNA would actually not be likely to cause changes to DNA because of the small amount, the brief life-span, and lack of mobility into cells. However, the bio-engineered versions (mRNA vaccines) are designed to evade detection by the immune system and have a lipid coating to easily pass into cells, are present in extraordinary numbers (10^14) and perhaps be targeted to cells which increases the risk by a lot, so the integration risk due to reverse transcription becomes significant, but how significant requires additional investigation.
This whole situation reminds me of the atomic bomb scientists.
The night before the first A bomb test the scientists were up all night debating just how dangerous the bomb would be to the atmosphere.
Some of them had run some numbers and were convinced it would ignite the entire Earth’s atmosphere and kill everyone on the planet.
But—the consensus was that more research was needed so might as well test it.
Boom!
Science, yo.