From what I understand of it, I admire Buddhism more than I do the monotheistic religions. There are different kinds of Buddhism, though, so we might be talking past each other. The kind of early Buddhism I've been focusing on recently says there's no such thing as the self, so "self-compassion" could hardly be crucial to that school.

What I said about the role of Buddhist compassion is just my opinion. If it's "inaccurate," that would assume a rather doctrinaire view of this therapy. I don't get so bogged down in the details here. As you say, the essence of Buddhism is the therapy itself, which is supposed to change the mindset and to free a person from unnecessary, foolish suffering.

Why would a buddha or a bodhisattva be compassionate towards others? As best as I can tell, it's because they've been freed from the illusions of egoism. They're not selfish because selfishness is based on misleading impressions that you can train yourself to see through and to lose. If you're no longer egoistic, you'd no longer have a reason to favour yourself at the expense of others. So why not help others?

But as I said, I suspect there's also an underlying humanistic sentiment here. Again, that's just my interpretation, or it's a hypothesis I'm exploring. Buddhism intrigues me, and I'm still learning about it.