It would be my impression that it would have been impractical to have tried to repair a defective musket while in the line anyway.

I would think that the man with the malfunction would either fall out of the line and go to the Sargeant where they would then make what repairs that they could. Or he would exchange weapons with the Sargeant and return to the line while the sargeant extracted his tool and made the repairs behind the lines.

That, to me, would be the simplest solution. A situation which I have actually had happen during a reenactment. In that case it was not a problem extracting my Sargeant's Tool from my haversack while kneeling behind the battleline and replacing a plugged nipple on the musket in question and returning the trooper to the ranks. After all the 3rd and 4th Sargeants were supposed to be position behind the lines as file closers anyway.

In actuality there wouldn't have been very many "repairs" that could have been made on the battlefield anyway in my opinion. Changing or unplug a fouled nipple or pull a bullet from a damp charge would have been about it.

Anything else would have required replacing parts which were probably not available to soldiers during the battle and any such repair would have to been returned to the Ordinance sargeant for repairs. So I don't see the necessity of carrying the enfield tool on your belt for quick access.