Nor did they write, "the right of citizens to keep and bear arms", which is what you are implying it means.
Ah, so you agree that there is meaning in the fact that they wrote “the right of the people” instead of “the right of militia members”!
Yes, they did not use the term “citizens” - perhaps because the right extends beyond formal citizenship to humans in general. In protecting a right very broadly, perhaps they considered the right should be protected for not only citizens, but also for those who are not formally citizens but nonetheless are part of the community which forms this nation (such as Native Americans and permanent resident aliens (such as my wife)). In trying to write a consice statement enunciating protection of a right, it behooved them to not get caught up in trying to precisely define a line which they had no intention of drawing: suffice to say “the right of the people” to construe the intention that everyone has that right.
As Hamilton noted the desirablility of having “the people at large” armed, others wrote similarly, and nowhere did anyone argue for _limiting_ the broad coverage of the term “the people”, it is safe to presume that “the people” meant everyone*.
(* - insofar as you may roll out examples of people who should not be armed, the issue of “reasonable restrictions” comes back to those who are demonstrably/ajudicated/sworn a danger to society, at which point others are free to exercise their 2nd Amendment right to disarm them.)