The emails are sent due to the record assigment, and go out to the user that the record is assigned to.
What you did with the other user is just a security groups setting that let’s you condition visibility of the Lead. But it changes nothing about the fact that the record is assigned to a single user, and it’s not the “lead viewer”.
I would do this with workflows - you can make a Workflow for new leads that sends an email to someone else, apart from the assignment email.
Or alternatively, in some specific condition, have the Workflow detect a change in the record and assign it to the “lead viewer” user, and the automatic assignment notification will then go out to him.