A Survey Designer user publishes a Survey through Multichannel engine (MC Engine). The Survey is in published status but the notification email is not sent to the user.
This is expected behavior. If a Survey is published through MC Engine, notification email is not generated and sent to the user.
This is working as designed. A notification email is only sent to the user if the Survey is published through the VEEVA_BATCH_SURVEY_PUBLISH Apex Class.
CRM Help Documentation: Publishing a Survey