I think nylon brushes, warm soapy water are the way to go
I like that you have your grandfather's toolbox. I have my dad's Kennedy from when he worked as a machinist when I was younger. Not to old, not wood, but solid and holds history.
That website had me going to.. I didn't know about the brand Best Made before so I was taken back with their video as well. I wasn't sure if the plunger video was a spoof on the axe video that I thought was a spoof on the hipster Pinterest axe scene.
Kudos to Best Made for liberating cash from young urban professionals wanting a good axe or hatchet

I like the council tools axes, from what I have viewed online.
I can't afford or justify to my better half buying just one axe at those prices, so I am immediately not in their demographic. However if I had a budget for that I would spend it at estate sales and flea markets anyway.