I think it's only natural to want to care in this situation, and also to want your caring to mean something to your counselor. And (to me at least) it can feel like the worst sort of rejection when it seemingly doesn't.

One possible interpretation of the mixed messages you're getting is that she's over-compensating for her previous slip-up. Even if she's not a therapist, counseling sessions should be for you and about your stuff rather than the counselor's. And she probably knows this. So she might overshare and not be aware (due to grief/related issues? or just the way she is?) until you show concern, then think 'Oops, did I really tell all that to the client? I better not share any more!'

BTW, I'm not sure it's safe to assume that when people share seemingly important and personal info then that means they want you to care. They might just need to vent and then forget about it. Not saying your counselor does this (would be quite inconsiderate if she did), but I think it's a thing in general.

I think it's a good idea not to ask about the funeral etc., but might be worth eventually trying to talk about how you feel, or how this is confusing for you. Unless you think it's likely to do more harm than leaving something this important unsaid?