Reacher’s idiosyncrasy of traveling with only a toothbrush is most likely a homage to the old Junior Walker and the All-Stars blues song (I’m a) Roadrunner, which is a great song in its many versions.
“Money, who needs it
Let me live a life free and easy
Put a toothbrush in my hand
Let me be a traveling man”
Along with Reacher’s references to old blues artist like Bobby Blue Bland, very cool aspect to the show