I am a law student who is nearly done with school.
Unfortunately, I was not able to sway my family. They felt that the potential danger to the average citizen was too great and that having full autos severely limited was a good thing.
Your story illustrates a few lessons.
Discussing law with laymen
Most often, laymen regard law as a political matter. This is to say that they look for the law to support their politically reached conclusions. This is not a matter of malevolence or corruption. The complexity of law discourages people from developing coherent and consistent views of it. It is also a fairly normal human response to see little value in liberties we choose not to exercise. Most people will not have a visceral reaction to laws that outlaw sagging trousers because normal people don't engage in the practice.
In the face of your recitation of "shall not be infringed" to your family, they knew two things. 1) they do not agree with the language and 2) enough people involved in the legal system have disagreed with it enough at some time and places to have infringed it. Since their position isn't built on COTUS language, your recitation can't address the basis of their position.
The influence of culture on law
I concur in your observation that we regard guns differently as a consequence of how we regulate them. The consequent strangeness of the regulated item makes it politically easier to regulate.
One of the duties you will have as an attorney is to read the law with integrity. This will often place you in a politically unpopular position. Many of your brethren will adopt a vision of the law in which the clear intent of basic law can be explained away for political ends. You will choose how you explain your own fidelity to law.
The dangers of result oriented jurisprudence
Many courts have decided first what result they want, then tortured law, language and reason to make that result happen. This is not just a problem in constitutional law; it makes a hash of the law of many states as well.
Edited to add -
I think you are correct to question whether there is any rational basis for restricting or taxing differently a rifle depending on whether it shoots automatically or semi-automatically. I also see lots of harm in making sound suppressors more expensive and rare.