Some overlay elements like dropdowns or flyout menus automatically hide/remove themselves when the focus is lost. Whenever you switch to the DevTools the page is not focused anymore and the element gets hidden.

The Chrome DevTools have a feature to emulate page focus. Open the DevTools, press Ctrl+Shift+P (or Cmd+Shift+P on macOS) and search for "Emulate a focused page".

Emulate a focused page in DevTools