But I see a collection of kits at https://www.amazon.com/Tire-Repair-Kits/b?ie=UTF8&node=15706711

The stretch action today is due to the taper of the rubber plug. Never saw one with a tool that grabbed the plug as you describe.

-> About sidewalls. Here about 99.99% of consumer tires should not be repaired with this if the damage is in the sidewall and depending on the brand, too close to the sidewall.

https://www.tirerack.com/tires/tiretech/techpage.jsp?techid=187 gives a quick rundown of this danger, do not repair area.