>I think running eslint --fix will even change your "let"s to "const"s where appropriate, but don't quote me on that.

Yep, this is the relevant rule - http://eslint.org/docs/rules/no-var

Though it just replaces all var with let. I expected it to intelligently use either const or let depending on how the variable is used ¯\_(ツ)_/¯