I either agree with several of the things you say, except in red... no, it doesn't. The mRNA vaccines do not enter the cell nucleus and do not interact with the cell's DNA.
Also, calling mRNA vaccines "gene modification therapy" that "messes with people's DNA" is just factually false, and calling that out is not nit picking. IMO, it is false disinformation much akin to when MSM/TPTB call AR's "assault rifle/machine gun/weapons of war"... and, yeah, IMO both are fear-monge
Of course there are other opinions...
COVID-19 vaccines are not conventional vaccines made with live or attenuated viruses. They're actually gene therapies. The Pfizer and Moderna vaccines are made with lipid nanoparticles that contain polyethylene glycol (PEG)8 and messenger RNA (mRNA).
mRNA are snippets of genetic code that carries instructions for cells to produce proteins. The definition of genetic is "relating to genes" and genes contain instructional code that tell the body what proteins to make. Therapy is the medical treatment of disease, so mRNA vaccines are very clearly gene therapy.
As noted by David Martin, Ph.D., Moderna's SEC filings9 specify and stress that the FDA considers its technology a "gene therapy technology," originally intended for cancer treatment. Its mechanism of action also confirms it to be gene therapy. The mRNA gene therapies currently being misleadingly marketed as "vaccines" turn your cells into bioreactors that churn out viral proteins to incite an immune response, and there's no off-switch.10