Outside of privacy, I could imagine it might be beneficial to structure a set of companies as a primary company and a bunch of shell companies to separate accounting and legal matters. This way if a sub-company is in a dangerous industry or has large liabilities in its normal business that you would want the primary company to be insulated from that. IANAL so I don't know if this is a use case, there are probably ways to do this w/o privacy.
As an example, it’s extremely common for financial companies of all sorts to create a company to hold an asset or group of assets. This can be done for all manner of reasons, including difference in beneficial ownership.
A VC fund, PE fund, etc will often have a “manager” (the possibly big-name company that operates it) and a bunch of limited partners for that specific fund. If you own a piece of a BlackRock fund, for example, you don’t own BlackRock itself. And the corporate structure reflects this.
