I feel like a private/closed blockchain would defeat the purpose. I want a future where a public blockchain would act a repository of everything that has happen

I feel like a private/closed blockchain would defeat the purpose. I want a future where a public blockchain would act a repository of everything that has happened to the final product throughout its history, on top of which applications would be developped, relevant for each step of the chain. The producer A that would input certification of origin, the manufacturer B, and the end consumer C would need very different services, web, app, industrial applications, ERP integrations and so on. To express it in an analogy with current services, I would like the blockchain to be a database, exposing a universal API. I don't really see a private blockchain allowing this, but I may be wrong. And indeed there will be the thorny authentication issue... I don't know the solution, really !