But a user wants to search the Start menu, not the taskbar (and who knows what else, Cortana). That's not an obvious way to communicate equivalence.

Step 1. In the interest of "simplicity" and hiding details from users, make more and more functionality only available through the search box.

Step 2. Move the search box to an annoying position on the task bar.

Step 3. Give users the ability to hide the search box.

If you click the windows key on the keyboard or the task bar, until you start typing, the search field is hidden.