Hmmmm … dunno.
When you describe it that way, I think of ‘sympathy’ rather than ‘compassion’.
In order to be empathetic, I need to be able to put myself in your shoes, yes, but that doesn’t mean I need feel any sympathy with you nor be compassionate — I could feel empathetic because I’ve made the same mistakes for the same reasons (perhaps I let my business and marriage collapse for some reason) but feel no sympathy because I blame myself for them for whatever reason (perhaps I was an alcoholic, for example) and have none for myself as a result.
Equally, whilst I may have no sympathy, I might, nevertheless be compassionate and treat you less harshly than I would myself because I am not you and, therefore, feel that I should treat you more gently, because I don’t know your circumstances as well as I might like to think — I may judge you without being judgemental of you.
Alternatively, I might feel empathy but no sympathy and show you no compassion either but treat you with so called ‘tough love’ of the kind I feel I deserve myself.
Human psychology is complex and that’s why we have the three related, but different, words to discuss the matter with, because one, or even two, simply aren’t enough to differentiate how we think/feel (empathy/sympathy) and how we act as a result (with or without compassion).
But that’s just my opinion of course.