That's not how it works. You need the signed cards to hold an election. Employers can, seeing a large majority publically willing to sign the cards before they have any solid protections, choose to forgo the election on the basis that a majority of employees were willing to sign.

If the only reason WMF had to request a formal election was to allow secret ballot, they would not be hiring union busting law firms. The point of such firms is to legally and illegally discourage union membership - the penalties for doing so illegally are comically small.