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.
I used to be an Auto Tech about 15 years ago. I worked for a guy that had a very special tire plug insertion tool. This tool only worked with the “Bow tie” type of plugs. This tool had the bottom part exactly like any other insertion tool but the top/handle part had a mechanism that slid up and down on the insertion stem and also grabbed and stretched the plug before insertion. I cannot remember the name of the tool manufacturer or where he got that darn thing from to save my life. I have scoured the interwebs for even a mention or picture of this elusive tool. I am hoping my description will trigger somebody to recall anything more than a vague description like I have. Please help! Thank you in advance for any information you may have!