[Auth] NFC - AuthRecaptchaVerifier.swift

[ncooke3]

This class is confusing, and I want to refactor it to get Swift 6 conformance. Going to do it gradually to make it easier to review and less prone to mistakes.

This PR:

• marks private functions as private
• moves private functions to bottom of implementation
• removes unused static var recaptchaClient

#no-changelog

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