Actually, Companies House does require that private companies ("limited by shares") file an annual statement listing all their shareholders. It's the "AR01" form and is available for free under the Filing History tab on a company's page - e.g. this is one of the companies mentioned in the article: https://beta.companieshouse.gov.uk/company/05292775/filing-h...

This doesn't force the disclosure of beneficial ownership though. Shares can be held by nominees without that being disclosed on the register. If the shareholder is an offshore company that can also be difficult to trace.

Another fun fact: it's possible to work out the exact most recent valuation of a UK-registered startup by reading its SH01 filing (when it issues new shares). I'm always surprised that the tech press doesn't do this as a matter of course given how excited they get about funding announcements...