First of all, that does not communicate all that the passed version does. While it still protects the individuals' rights to keep and bear arms, it doesn't stress something that was very important--the importance of the militia.
And specifically, it's there because of the history of how the amendment came to be passed. The wording comes over nearly verbatim (in James Madison's original proposal for the amendment) from George Mason's Virginia Declaration of Rights, though with additional clauses dropped. With the additional clauses dropped, the form is more awkward, but it's obvious that the dependent clause remains as it was a very important point--and one of the key points from the original--that the militia is the best defense of the people against a standing army or the government.
In fact, earlier versions explicitly stated what George Mason himself had said--that the militia was the whole body of the people--but the wording was streamlined and the extraneous definition of militia was dropped.
Similarly, the Senate felt so strongly about the need to emphasize the importance of the militia as a check, it changed "the best security of a free state" to "NECESSARY for a free state." On the other hand, that same day, a proposal to add "for the common defence" after "bear arms" was shot down.
But really...most of all, I think...the opening clause of the Second Amendment is necessary because it is dealing with a topic that is mentioned in Article I, Section 8, and without clarification, someone could try to claim that militia equipment would be exempt from the Second Amendment. The opening clause makes it quite clear that although the Congress is given the power to play with the militia, the Second Amendment prevents it from defanging the militia by removing arms from the people or making the militia a tool of Congress to the detriment of the body of the people.
I ask, sir, what is the militia? It is the whole people, except for a few public officials." — George Mason, in Debates in Virginia Convention on Ratification of the Constitution; June 16, 1788
I believe that definition applies to the entire amendment.
"George Mason himself had said--that the militia was the whole body of the people"
As you correctly quoted at the end of your post, George Mason said that the militia is "the whole people". "The whole people" was another way of saying "the people".