You're excluding a key point - the policy often benefits one party at the cost of another. You mentioned immigration and that's a great example of this sort of pathological empathy that has infected the left.
There's a cheap and fleeting sense of virtue attained when you champion illegal immigration and decry deportations. You post photos of mothers and their children crying at the border because the human trafficking organisations are having a hard time getting them across nowadays. But it's important to remember the negative pressure illegal immigrants place on wages and why there's a gross cabal of large corporations, lobbies, and affiliated NGOs, who virtue signal immigration as a means to lower their labor costs. It's important to remember the entire pipeline of illegal immigrants is owned and operated by extremely violent cartels - humans are now their most valuable product. Your desperate craving for that high of in-group acceptance is propping this up.
It's not that you're empathetic - you just don't care about the negatively impacted party. Nothing new under the sun.