Could you check the "Sending statistics" in your AWS SES dashboard and see if there was any activity there? Also, is that email the one you configured as the sender or receiver? Please note that SES limits are set based on reputation, so having a high bounce rate (e.g. sending emails to non-existing addresses) could cause AWS to heavily restrict your ability to send emails. You could also check this in the "Reputation Dashboard" insides the AWS SES control panel.
Also, just to confirm that the emailing functionality in Weblate works correctly and it's not a matter of configuration for SES, could you try configuring an alternative SMTP provider, e.g. Gmail, as described here?