Lets be fair, that is a comedy show who has an extremely liberal intended audience. I am not saying the daily show doesn't occasionally produce valuable journalism, but I have never see then produce anything that did anything but mock a conservative position. Lets get informed through experts, not comedians.
Now the base finding here is probably essentially correct, but the methods to produce it were extremely different from reality. For one thing mass shooter simulators are inherently extremely unrealistic, because you know before hand that there will be a mass shooter, and generally what the parameters will be. This is the single biggest problem when this actually happens, and the hardest thing to simulate. I know from personal experience that when guns start going off there is always a massive amount of confusion as to what is happening an where. The more you know about the situation, the faster and more effectively you can react. In a simulator you have a lot of information, in reality you have little to none.
See what I just said above for why arming even 20% of teachers is a really bad idea. There are about 3.2 million full time teachers in the US, so we are talking about adding 640,000 guns to schools. It is far easier to train people how to clear rooms or conduct SWAT style raids then it is to react to an active shooter, due to the reasons I talked about earlier in this post. This is more people then the entire US Army, which is an organization that devotes about 80% of its time to training, and I can personally vouch for the fact that at least 70% of them are still completely incompetent when it comes to handling weapons (Granted, most of the training isn't weapons training, but weapons training is required for everyone at least twice a year).
As for panic buttons, the concept sounds ok, but the logistics sounds like a nightmare. The money spent on that would be much better spent doing better funding for low income school districts, which I guarantee you would have a far bigger impact on gun violence then bulletproofing everything.