Correct. It does not enter the nucleus and mRNA disintegrates very quickly...it does not hang around.
I think some people are conflating this with the Astrazeneca vaccine which is not approved for use in the US...that one actually does send things into the nuclear membrane. That gives me pause, as that could influence the cell permanently - I don’t know that science on that, but what if cell while having the material reproduces through mitosis? What would the long term effect of that be?
That is...quite unlikely. Typically when a virus invades a cell it starts to replicate inside the host until the host bursts open. That is, the host gets physically destroyed. But even if it weren't by the replicating virus, the presence of the virus inside the host would cause the host to sprout foreign (ie abnormal) proteins on the cell surface. The immune system would then identity these infected cells from their surface proteins and destroy them. That's typically how you recover from a cold.
But with the AstraZeneca vaccine there is a catch. The viral vector used to deliver the spike protein DNA into the host is a chimpanzee virus - specially engineered to NOT replicate when inside a host. It's job is to make the host sprout the spike proteins on its surface and let the immune system react to it. But if it's not going to replicate inside the host, I guess it's only up to the immune system to destroy the infected cell. The immune system will almost certainly (>99.9%) kill it the same way it's being primed to kill the Covid virus from a possible future infection. But what if it doesn't ? Does that mean the chimpanzee virus infected cell survives ? I know it's extremely unlikely but that's the part I am not clear on. Will have to do some research.