When you push a fresh bit of wire into the slotted MTA wire receptacle, the slot displaces the insulation from two sides and then eats into the wire from those sides. Maybe that's where the "cold welding" action happens, and why you need to pay particular attention to the wire gauge (in your case that shouldn't be a problem.) The thing is that the tool (I have the little dinky 35 buck hand tool) has teeth that allow you to push the wire by two ends over the U-shaped slot/receptacle, but without pushing the slot itself. I'm not sure how you can achieve that effect with an unodified screwdriver. Even with the tool MTA connections are a bit so-so IMHO.
If you have the hand tool, pull out the offending wire, trim it so you get a fresh start and then you should be able to push it back in with the tool, I've done so a couple of times and it seems OK.
If you don't you might make a decent MTA tool by taking an old screwdriver and filing a little slot in it.
I wouldn't personally trust wire that has been just pushed in with a screwdriver.