You *must* have an email address in the user profile if you are the site administrator. If there is someone else managing your site as an administrator, and you are logged in at a lower user level, then maybe you can remove your email address from the profile, but be aware that if you ever run into problems logging in, then you will not be able to use the recover password option on the login page if your email is not linked the user account.
What do I do if I want the person who manages the website to receive the core update email notifications?
If you add that code to the functions.php file it will stop *all users* from receiving update messages, including the person you have managing your site. If you don’t add the code, but *do* use this person’s email address in the admin profile account, then this person will receive all WordPress-related notification. Note that this person will then be the acting administrator of the site, with the ability to create and delete all other users (i.e. you should have a high level of trust in him/her).