Saying more basic tires is dangerous is somewhat like saying a BRZ or Sport-Compact is dangerous.. With this mentality you are basically saying that you should put the biggest extreme summer tire on your VW Beetle that you can so you don't get "caught out" and sure, I can understand that philosophy.
If you get caught out doing something stupid or put into a situation where you need maximum mechanical grip and are going to be pushing a full lateral load, yeah the best tire you can buy (RE-71R) is going to save your ***. An example would be If you go over a crest and didn't see a tight bend was coming up and you are driving to fast for the road or conditions.
Most bad accidents though are going to be traffic accidents, and swerving to avoid uncoming traffic or having to slam on your brakes isn't going to make that large of a difference because in most of those quick-react situations you aren't pushing the tires envelope that far.
I know from owning an older Z and now a 370, is that if you try to get close to the limit on public roads, it is much more risky. If you want to feel the 370 get close to the edge of grip on an empty country road even with some hard slower bends with the stock tires, you are going to be murdering the speed limit every where and if you know the limits of your machine, you will be safer in a car that has skinnier and less grippy tires when you do find and go over that limit.
If you don't know how to find the limit of your car regardless of how much tread is on them, well then I do hope you have the grippiest tires you can find.
Also, as per the Chris Harris video, you could always add more power to decrease the effectiveness of your tires when on throttle!