Enable and fix `@typescript-eslint/explicit-member-accessibility`

[jtbandes]

User-Facing Changes None

Description Rationale: With public being the default accessibility level, TypeScript makes it easy to accidentally expose methods and members that were intended to be internal to other files/classes. In a large codebase this can lead to unintentional entangling of components over time. By making an explicit accessibility level required, we encourage developers to think about the accessibility level of each method/field, which will hopefully steer them towards more private or #-private methods/fields.