No, we can do that. The government can merely assume the power, as governments so often do.
Paper is not a defense. A "Bill of Rights" containing only a single right would be of zero value.
What is a defense is the bundling of recognized rights, as in the Bill of Rights. It's like an alliance: an assault on any one right is an assault on them all. If a large majority of people would rally behind a single right (maybe freedom of speech?), then the right is not seriously in danger (for now). The rights which are in danger need to be bundled so that the whims of a particular time and place do not permanently sacrifice a right.
When a particular right is borderline unpopular -- perhaps the right to keep and bear arms -- that provides a lever to pry away all of our rights, and we must prevent that. When you look at the wave of gun laws being proposed and passed, you'll see how obvious it is that they are an assault on the entire Bill of Rights (for instance "red flag" laws are a clear violation of due process).
The Gun Safety movement is a rerun of the Drug War. A bunch of reasonable-sounding people concerned about their community used as pawns to push a very different agenda. In the case of the Drug War, it was violating the 4th, 5th, 9th and 10th Amendments to oppress political enemies (hippies and blacks). In the case of the Gun Safety movement, it's violating the 1st, 2nd, and 5th Amendments to oppress political enemies (rural/Christian/whatever).
The reason we end up with ridiculous laws -- like harsher punishments for crack than powder; or laws restricting foldable stocks on rifles[1] -- is because those laws are not ridiculous when you understand the real agenda.
[1] If the goal is to reduce murders, then rifles are an absurd place to start. Knives are a much more common murder weapon. So are "Hands, fists, feet, etc.". Even in Texas, where there are few rifle laws. https://ucr.fbi.gov/crime-in-the-u.s/2018/crime-in-the-u.s.-...