You seem to acknowledge that the good guy with a gun, most likely, cannot kill or fully disable the shooter in this situation.
zerofoo said nothing of the sort. He said that killing or fully disabling the shooter isn't necessary, not that it's not possible, or even unlikely.
The history of mass shooting violence in the US bears out both zerofoo's point that killing or disabling the shooter often isn't necessary, since mass shooters tend to suicide as soon as they meet armed resistance, and even those who don't are clearly going to have to shift their focus from mass murder to self-defense or be an easy target. There are plenty of cases in which shooters have been killed, disabled or otherwise stopped by citizens, though they tend to get less press for the simple reason that fewer people die.
So now you have to calculate, how many people could potentially be saved in that scenario, versus how many people would be killed if handguns are more widespread. If 5% of the populace is armed now, what happens to the death by gun rates when its 10%, 20%, 75%, 100%?
Interestingly, the US has conducted this experiment over the last 40 years or so, as the number of concealed carry permit holders went from basically zero to up to 15% in some areas of the country. What happened? Not much. Violence declined, and there is some weak statistical evidence that it declined faster in areas where more people began carrying guns on a daily basis. There is no evidence that violence increased.
Will the number of random shootings go down by allowing anyone, including the mentally ill, to have easier access to guns without waiting periods?
I suspect it won't change at all. What would reduce the number of shootings is removing all of the guns, but that is impossible.
P.S. Several school shootings (like Columbine) have taken place at schools with an armed police presence.
This is something of an unrelated point, but I think it's worth noting that police did not respond quickly at Columbine High School, and that the experience dramatically changed police doctrine for responding to active shooters, across the nation. After-action analyses showed that as soon as the shooters faced aggressive armed response, they killed themselves, but that happened many minutes, and several deaths, later than it could have. At the time standard procedure was to cordon the area, isolate the shooters and wait for enough backup to arrive -- preferably SWAT -- that the police could enter in overwhelming force. But Columbine changed that, and most police departments now train their officers that if they have good reason to believe it's a lone shooter then the very first officers on the scene should enter immediately.
Immediate entry, without overwhelming force or special weapons, seems like it significantly increases the risk to the officers, but in practice it doesn't, much. And, of course, waiting tends to give shooters time and space to rack up massive body counts.