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awkwardly-phrased amendmentIt's only awkwardly-phrased if you try to read it with 2017 Standard Coastal American English and without the context of the complete document (as it is supposed to be read, evidenced by the rather broad definition of speech and the press applied to the first amendment), the intent of the drafters, and a deep understanding of the writing style and cultural influences on interpretation which affected the drafters' choice of words.
> could easily be construed differently than the NRA does today
The NRA construes the second amendment to the U.S. constitution as the Supreme Court interprets it. If you have a disagreement with the Supreme Court's interpretation and standards regarding the second amendment to the U.S. constitution, state them in terms which thoroughly challenge the painstaking review of evidence which informed their interpretation. I recommend going to the wonderful folks at Oyez, they have recordings (the official transcripts have glaring errors, including literal binary flips of meaning) of hearings and briefs for you to review and couch your disagreement in.
> designed at a time when "arms" meant something vastly more innocent than today.
This is false, don't rely on pundits to understand and accurately convey a history of the design and manufacture of arms. There were semi-automatic firearms (handguns and rifles) of arbitrary magazine capacity in manufacture at least 40 years prior (and if you include the Kalthoff repeater [which may have not been commonly known in the colonies, and requires two actions to fire], 155 or more years prior) to the drafting of the second amendment (December 1791). Furthermore, the amendment actually makes no attempt to define the type of arms covered, not even defining that they be firearms or specifically rifles or handguns or guns (cannons) or shotguns.
> So I personally would rather we focus on natural rights that reflect today's technology, rather than artificial ones from the 1790s.
The right to keep and bear arms is a natural one, because the purpose is self defense, and the common defense. Humans are defined largely by their capacity to design, build, and/or use tools, so it is obvious that access to the tools which have become essential to the natural right to defend your person would themselves warrant the same recognition.
It is a natural right to use appropriate and proportionate means to defend your property (including your life), and to retain those means in advance of such an occasion.