No, but they would probably become less deadly.
(But, without a gun, people might not be so emboldened as to physically attack someone at close range. Violence might go down.)
Around the time of one of the mass shootings in the past 5 years, my conservative relatives were posting on facebook a story about a mass-stabbing. They asked rhetorically, "Why aren't we calling for a ban on knives?" Well, the question sort of answered itself-- no one had died from the attacks.
A tautology is "a statement that is true by necessity or by virtue of its logical form."
That statement may be "obvious" but it is not a tautology.
"More x are used when there are more x" is not a tautology. It is often true, but it is not a logical necessity.
Evaluations after the reforms suggest that they saved lives. A study by Andrew Leigh of Australian National University and Christine Neill of Wilfrid Laurier University estimated that buying back 3,500 guns per 100,000 people led to a statistically significant drop in firearm suicides — 74 percent, in fact, with no parallel increase in non-firearm suicides. ...
The results on homicides were a little less clear. Leigh and Neill found that the buyback resulted in a 35 to 50 percent decline in the gun homicide rate, but because of the low number of homicides in Australia normally, this change wasn't statistically significant
I don't have the numbers for Austrialia, but from the same article:
Nearly two-thirds of gun deaths in the US are suicides
If the proportion in Autralia are similar, reducing gun suicides, if not homicides, could save many lives.
This is of course a tautology. If you were to magically remove that tool, do you really think Americans would become more peaceful???
As for your other numbers, no one denies we're the top in per-capita civilian gun ownership. Good laws + wealth lets us swamp Serbia and Yemen, after which comes ... Switzerland. Known extremely violent country (well, they were notorious so back in the days of pike bearing infantry).
Not sure the U.K. is that useful, they've been suppressing civilian gun ownership for a century, and they're on a small island.
"Perhaps the single most supported contention in all of gun research is that more guns mean more gun deaths."
I'm sure there's a Clint Eastwood quote about it mattering who gets shot. More to the point, to the extent criminals are killed by civilians, what's the general problem?