Yes, there is the "initial secret" problem. There are a variety of different ways Vault handles that (look up the auth plugin docs), but at my org we use AWS IAM auth. So, app servers authenticate via their IAM instance profile (provided/managed by AWS) while developers assume a specific IAM role and then authenticate via that (which is how we enforce MFA for Vault without paying the crazy enterprise pricing).

Note that with AWS IAM auth, AWS is a trusted third party, and accounts with high-powered IAM access (think AWS admins) end up having a great deal of authority in Vault, too. But for us, at least, these assumptions are reasonable.