I was hoping for some easier way to close off a couple web part on
MySite without messing with the template.

Is this just for you, or are you trying to make the change for all users?

Also, are you trying to update the public profile  page or the private My Site
page? There are two views for My Site.

One clarification... users  do not control the "My Profile" layout,
administrators do. After re-reading your post, I realized that you may
not realize this and may not be an administrator. All users have the
same "My Profile." However, users can update their "My Home."