We would like to change what is displayed on the person.aspx page for
all users.

It does not look like that adding/removing webparts on the person.aspx
page is enough. For example, the default contact information is not in
a webpart.

What is the best/right way to accomplish this? Right now, we created a
person1.aspx page (a copy of persona.aspx) and made changes to the new
page.

But on SharePoint, there are many places you can go to the person.aspx
page, how do I make sure that all the references to person.aspx is
changed to person1.aspx?