tl;dr - the point of a weapon in this situation is to project and multiply force, not to kill. Firearms are the best means of doing so.
> You think you need a gun in order to kill a would-be assailant. The OP says he used a knife to scare them off. I know which I prefer.
I guess the point I'm trying to make and I should have phrased this more clearly, is that the primary purpose of having a weapon in this situation (either as a criminal or in the case of home defence) isn't to kill someone - it's to project force. Guns and knives are both force multipliers, it's just that a gun is much more efficient and therefore a much better force multiplier. The point isn't to kill someone, it's to coerce them with the threat of force to do what you want them to.
OP was making that threat when he scared off his home intruder(s) by holding out a knife and 'leaving them an exit' - in essence he was saying "leave my home or I will use potentially lethal force on you." The situation doesn't fundamentally change if OP is armed with a knife or a firearm; it's just the strength of that threat is greatly increased.
Another way to look at this - if OP and his assailant are rough equals in terms of ability to do harm to each other, it doesn't matter if OP has a knife or a gun, as long as he has one or the other. Anything that amplifies his ability to enforce his threat of force should be enough to tip the balance of power is his favour (assuming a rational adversary). Guns only become necessary or useful when there's a disparity between the assailant and the now hypothetical OP. There's a moderately well known case in America where a Texas woman's home was being attacked by two men (it turned out they wanted to steal her husband's prescription painkillers). Her ability to coerce these men without a weapon was extremely limited. However, she had a firearm, and was able to stop the attack through the threat, and use, of force.
> I think the American style of "bringing a gun to a knife fight" just escalates everything.
I mean, you're right. The problem is that a) things have escalated, and b) de-escalating it would mean deep systemic changes to the American Constitution and the perceived balance of power between the State and the Citizen. The reasoning behind the 2nd Amendment is two fold - first it's to provide for common defense in the form of a State militia, which is largely a moot point in this country. The second, and this is a bit controversial, is to provide citizens with a means of resisting the Federal government in the event of tyranny. One would think this would be a moot point as well, in a country that spends an obscene amount of money on its armed forces, but it has moral significance - saying that at the last, the citizen is responsible for the protection of their own liberty, and preserving the means of that protection.