Pick up any (college level) biology textbook. It’ll explain the chemical processes involved. In short, mRNA is just a series of nitrogen bases attached to sugars that anchor them to a phosphate backbone.
Entry into the cell nucleus is controlled at the nuclear pores. Entry requires a transport mechanism. There’s plenty of transport proteins inside the nucleus, but their only capability is to take mRNA that’s already inside the nucleus and transported it to and out of the nuclear pores. So mRNA coming in has no known method and would be going in reverse. Maybe if we could reverse time.
If you could Harry Potter magic wand the mRNA into the nucleus somehow, it would be immediately transported back out by one of the transport proteins. If you could magic wand that away (well, you’d kill the cell quickly because you’d be shutting down a vital function, but let’s ignore that), then the mRNA doesn’t do anything in there. Only a specific enzyme has the ability to unwind DNA and mRNA has no effect on it. If mRNA could affect it, it would be doing so every time any protein is encoded from DNA. Same goes for cutting DNA, making inserts, etc. If mRNA could do that, it would be happening every time DNA transcription (the process of creating mRNA from DNA) takes place. In other words, thousands of times every second in every cell of your body.
If mRNA could cut and make inserts to DNA, your DNA would constantly be getting cut up and scrambled in every cell in your entire body and you would instantly die.
Here’s a short (less than 3 minutes) video that explains how the normal cell process works: https://www.youtube.com/watch?v=gG7uCskUOrA
The video is 6 years old, so it has nothing to do with the current mRNA vaccines or COVID-19. This is just basic human biology that’s been well understood for many decades. The concept of mRNA making changes to DNA is like the concept of a woman taking her toddler and somehow turning it back into an egg. We know that doesn’t happen because you’d need to run a whole slew of biological processes in reverse and break all kinds of laws of chemistry and physics.
I would strongly suggest that since any of those textbooks were published that a lot of progress has been made in the 'gain of function' research. And, since that research is so sensitive it hasn't made it into any of the textbooks that you make reference to.
I would also note that the reactions to the vaccine for many is severe onto death there is something that isn't being explained.