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You can get another tool, kinda looks like a real course rat-tail file. That cleans out the hole, & makes it a bit bigger (like you say to use a screwdriver).

Some of the tools are pretty soft. I had one that looked like your picture, but it bent when I pushed it into a hole that wasn't quite straight. YMMV...

I've had better luck with plugs that are made of rubber, instead of those that are made of rubber-soaked yarn.

If the hole is at the side, too close to the sidewall, you're SOL.

Better way is to use a patch inside the tire casing, but you have to take it off the rim. I've seen patches with built-in plugs where you pull them thru from the inside towards the outside till the patch is laying flat against the inside of the tire.