"The distinction is that the 4th amendment limits government power to intrude on our lives by forcing them to clear a certain burden of proof before seizing our property and information."
I think you're interpreting the 4th Amendment kind of like the 5th Amendment prohibition of depriving someone of life, liberty, or property without due process of law.
The part of the 4th Amendment that's interesting to this discussion is the prohibition on unreasonable searches. You haven't really offered any reason that should exist (information can't be seized, by the way, because the original owner still has it).
"I believe I have carefully evaluated whether my reaction is emotional. Frankly, I'm a bit offended..."
I got you confused with the person to whom I originally replied, who said: "I certainly would be glad to know that none are in my (crime-free) neighborhood." That's clearly an emotional reaction, but I apologize for attributing it to you.
The only rationale I can imagine for restricting gun rights is to preserve some proportion of lives (or maybe an absolute quantity). I would still like to see your argument framed that way. It's unreasonable for me to ask for a specific number, but it seems like you should have some order-of-magnitude percentage in mind -- e.g. A% of the population were violently killed last year; restricting gun rights will reduce that to B% in Y years, thereby saving (A-B)% of the population per year.
The problem that this line of argument faces, of course, is that the percentage of people killed by guns is already quite low. And more than half of those are suicides, which probably wouldn't be prevented by gun restrictions. Some other fraction would still be killed by other means. And, unless Y is a very large number, then you still may have many gun deaths simply because there are still so many guns around.
When you actually work out the numbers will all of these things accounted for in a realistic way, I suspect that (A-B)% will be a small number (or negative), and Y will be a large number. In other words, a small (or negative) benefit after a lot of waiting (and possibly bad consequences during that transition).
But it would still be perfectly rational for you to say that you believe it will save a single life, and that's enough of a benefit for you. You'd probably want to take some risk into account though, in case you're wrong and the life isn't actually saved.
"I think I'm turning the burden of proof around simply because the reasons against gun rights seem fairly obvious to me. Guns are tools built to kill, with no other use. They have very little in the way of useful legitimate uses."
You assume that the guns must be used to be useful, which is false. In fact, the presence of guns may reduce the frequency with which they are used[1]. Even police in a lot of areas have probably never fired their weapons at another human.
[1] I have no numbers to back this up, but again, my argument does not depend on numbers.