Unless integrase is coded for there’s no way for a piece of DNA to be integrated into normal host DNA. The cells used in this are abnormal eternal cancer cells, similar to the HUH7 cells used that “proved” mRNA could be incorporated into host DNA.

If you’ve ever had infection from any of the 9 herpes viruses then you’ve got foreign DNA in the nuclei of different cells, including your immune cells or your nerve cells. If you’ve ever had sixth disease as a child then there’s a 1% chance of it being attached to the telomeres of your DNA and hence passing it down the generations.

There’s a significant portion of human DNA that is viral in origin, part of which codes for the placenta, without which humans and other mammals would still be laying eggs.