Electricity doesn't travel down cores. It travels on the surface. Thus us why wire of #8 and larger is made of multiple conductors, so the electricity has a greater surface area pathway. Esd tools might be made of metal, but for all intents and purposes the outer diameter surface might as well be made of lead, it just doesn't conduct electricity well at all. So even if the component has a massive charge, and you have none, using Esd safe tools means when that tool in your hand makes contact with the component, there's little to no discharge. Instead, any discharge will be hitting you as soon as you clip the strap to your wrist.
The necessity for Esd safe tools comes with not having a linked path, like you aren't working on a mod mat that's strapped to you.
I often have to work on live circuits, many times working on unbreakered mains side wiring, and I do so with one hand in my pocket. Had a punk kid call me lazy for doing so, he wasn't impressed, even when I held a 120v live wire unshielded. No pathway. Reason for hand in pocket is to resist temptation and remove possibility of that arm touching the panel or a grounding/grounded wire and providing a pathway, up one arm-across my heart-down the other arm.
That's the purpose of the strap, provide a link making you + component a balanced load, with no pathway. If working on components with no way to break the path, regular metal tools provide a path and you get Esd. Esd safe tools keep the path broken, so no Esd.
Meaning if you only work on stuff that's on the mod mat, and you are strapped, you don't need Esd safe tools. The strap makes anything you touch the same charge as the components on the mat, = no path, no Esd.
Again I still don't understand the mechanisms by which the ESD tools work. And couldn't I just touch the metal part of a non-ESD tool to drain any charge built up on it when I'm connected to ground?
All that does is provide a possible path. It's not the tool that's the issue, it's the tool as an extension of your arm and body which has a charge. As soon as you touch anything with the tool, you provide a path and get Esd.