IMO the entire difference is whether the Government has, or should have, the responsibility to force the individual citizen from whom it derives its power to perform some action which government alleges would be of benefit. Clearly most of us would not support the idea that the government should force a healthy diet upon us. (See Pelosi-care)
However, the idea that common firearm ownership and training would benefit the common defense was obvious to the founders. Whether it is still a good idea is at least arguable.
If we the people decide that mandatory firearm ownership is a good idea (Swissman where are you?) then we the people need to provide the training as well.

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Ipsis Rebus Dictantitbus