Changing the Email provider to anything else than AWS Cognito or AWS SES – or suppressing the outgoing messages programmatically was not possible. The only possibility was to use a Lambda trigger to at least be able to modify the template of the message: Custom Message Lambda Trigger
Until probably a few weeks/months ago:
Custom sender Lambda Triggers
Amazon Cognito user pools provide two Lambda triggers CustomEmailSender and CustomSMSSender to enable third-party email and SMS notifications. You can Use your choice of SMS and email providers to send notifications to users from within your Lambda function code.
I did not find any release or change message on when they introduced those new triggers, but I am super happy I finally stumbled across this! I felt so much joy I almost cried.