I guess I'm in kinda the same situation as your person, minus the miracle cure stuff - my dad died pretty suddenly in December, I often talked shit about him while he was alive (he wasn't always the best - he wasn't always the worst, but he could be a real asshole sometimes, and I have years of therapy-level issues with both parents and a lot of stuff to do with my upbringing), and I categorically do not want to talk about it with the majority of the people in my life unless I'm bringing it up, which is unlikely. The person I talk about this with is my therapist, and I pay her for that service. I'm also generally a giver of we're-not-talking-about-this vibes when it comes to emotional stuff (including past & present mental illness stuff) - it's not just a grief reaction, but my grief has manifested in a way that's very consistent with who I was anyway. I haven't even told some friends who live overseas and I don't talk to often, because I didn't feel comfortable bringing it up, even though we were very close when we lived in the same city.

I think there's a fundamental conflict between what you want ("I feel like I have an obligation to communicate better") and what your friend wants ("they are still giving off the same vibes they always have") - to the extent that 90% of this is probably one of those "people are just really different in how they deal" things. I've had plenty of people who have said something to me about my loss because they felt personally obligated to do so (because that's how they deal with other people's loss/would want other people to deal with theirs), even though they know me well enough to know I'd find that awkward and unwelcome. They felt compelled to do it anyway, and it didn't help me.

Honestly, even though their intentions are good, it's felt like some people prioritised their comfort over mine, and that's not really comforting to me, even if it's meant to be. I also ended up on the receiving end of a lot of "let me tell you how terrible my grief about your dad is!", especially from his close friends/co-workers/neighbours, when I was in a totally transactional, businesslike, I-have-to-plan-and-execute-a-funeral, numb kind of a mode. That was really unhelpful.

Things that actually were helpful: friends who said something along the lines of "I'm sorry your dad died" once and once only when they heard, made it clear they still wanted to hang out, and then hung out with me as though things were totally normal and my dad had not just died. Friends who were totally cool with me making a bunch of grim dead dad jokes, and who joined in in ways that surprised and delighted me. Or at least who didn't make that one face most people make when you make a bunch of grim dead dad jokes. The people who made me feel weird were the ones who made a big, verbal, explicit thing about being there for me (some to the extent that, again, it was more about them than it was about me). The ones who I know are there for me by how they've acted (chill, fun, welcoming, no pressure) rather than what they've said or done are the people I feel the most comfortable around.

I realise I'm coming down pretty hard on the side of "work with the person, rather than whatever you're feeling about the situation", which I stand by, but there are some caveats. For one, I've definitely not been dealing with the grief in a healthy way - lot of suppression, not feeling a lot, a thought cycle that goes something like "huh > my dad died > that's weird > huh". I'm working on that in therapy. It's hard to know how much of what I feel about how I want other people to interact with me is being fuelled by the unhealthy grieving - maybe the best thing would be to let people in, but whatever's running the show right now sure as hell doesn't want that to happen.

Long story short - I would listen to what your friend is trying to communicate, not talk about it unless they initiate, not be weird if they do initiate and do regular friend stuff (and there's a huge and tangible difference for me between "regular friend stuff" and "I am trying to care for you as I know you are having a hard time" stuff - again, I appreciate the thought behind the latter, and can tolerate it, but it still feels a bit smothery) with them roughly as often as you would usually do regular friend stuff with them.