Metta Bhavana is more meant as a far reaching 'radiating kindness' across all beings on earth, as in to visualize literally everyone and wished them all well with deep warmth.
The purpose is more to love equally those who you would previously classify as loved ones versus people you don't like or don't care about. Which obviously is impossible for a normal human being unless you completely allow for some sort of 'death of ego'.
What you describe there in that we should love all equally is Christian thinking as per the Gospel, which I agree has many similarities with that Budhist meditation exercise but is not specific to a prayer for other people.
My whole point is that I find it interesting, and surprising, how western psychologists are taking concepts from eastern religion, completely ignoring the western religious tradition.
"He’s examined a technique known as ‘loving-kindness meditation’, for example, which is inspired by the Buddhist practice of Metta Bhavana. The practice involves contemplating people in your life – from friends and family to acquaintances and strangers – and cultivating good wishes and feelings of warmth for them."
This is regular Christian prayer. We Catholics pray for other people, every day.