its not the same problem, now anyone who wants to know who you are has to pierce the corporate vale. If the holding corporation is incorporated in a friendly jurisdiction, then you have extra big steps toe pierce the corporate vale (Malta, Kit Neives, Bermuda, etc)
A post box would be the best option IMO, as well as another contact number. Those combined are more expensive than just purchasing the domain with privacy protection included though...
If you're not asked to add whois privacy at the checkout in future, the registrar or domain owner probably don't allow it