AWS Cognito USER_PASSWORD_AUTH "Initiate Auth method not supported."

I had the same issue - resolved it by using AuthFlow ADMIN_NO_SRP_AUTH rather than USER_PASSWORD_AUTH per the example here.

In case of Serverless framework usage, the ALLOW_USER_PASSWORD_AUTH need to be added to the ExplicitAuthFlows node.

Resources:
  CognitoUserPool:
    Type: AWS::Cognito::UserPool
    Properties:
      # Generate a name based on the stage
      UserPoolName: ${self:provider.stage}-user-pool
      # Set email as an alias
      UsernameAttributes:
        - email
      AutoVerifiedAttributes:
        - email

  CognitoUserPoolClient:
    Type: AWS::Cognito::UserPoolClient
    Properties:
      # Generate an app client name based on the stage
      ClientName: ${self:provider.stage}-user-pool-client
      UserPoolId:
        Ref: CognitoUserPool
      ExplicitAuthFlows:
        - ALLOW_ADMIN_USER_PASSWORD_AUTH # See also: https://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/aws-resource-cognito-userpoolclient.html
        - ALLOW_USER_PASSWORD_AUTH
        - ALLOW_REFRESH_TOKEN_AUTH
        - ALLOW_USER_SRP_AUTH
      GenerateSecret: false

# Print out the Id of the User Pool that is created
Outputs:
  UserPoolId:
    Value:
      Ref: CognitoUserPool

  UserPoolClientId:
    Value:
      Ref: CognitoUserPoolClient

Under the App Clients section, did you enable the box for your user pool that says:

[x] Enable username-password (non-SRP) flow for app-based authentication (USER_PASSWORD_AUTH)