Why on any level, whould it help the grief to keep things a secret? Why turn an illness into something secret? I admire the way the family handled the death of the Norwegian princess' ex-husband some years back. They could have said he had a heart-attack or something, but openly told that he chose to end things himself. I am NOT suggesting that this happened in Jacks case!! I'm just saying that being open and let people show their love probably helps the ones who are grieving too (and it prevents speculations).
I don't read gossip -pages, don't follow "influencers", and hardly watch any reality-series. I don't think people should share absolutely eveything on social media. I do believe in openness and dialogue. To share emotions. We are all humans
The more popular you are, the more should you care about privacy. Grief for a normal human being isn't the same as for someone who has any popularity at all, even a relatively humble one like Jack's. He wasn't from the royal family, but still.
Usually, the public news of famous people's deaths arrives after a few days, exactly for families to survive the initial shock. Getting a bunch of messages, calls, emails, journalists, and possible scammers would be too much to handle.
Imho, if a person never declared it in the first place that any information about his live should be public, it should be kept secret by default.