They are doing it the correct way. In order to get union recognition you need to get enough people in the company to sign union cards. This is the normal way to create a union.

Companies argued that some people could be coerced and so the NLRB created a "two step" process in case the company protests the creation of the union. Basically if the company says "we don't think these cards were signed under proper circumstances" the NLRB would come in and hold a secret ballot.

That is basically what's happening here, and "wikipedia's side" is actually them reading a statement written by their lawyers in order to activate the NLRB balloting process.

The real reason why the companies like to invoke the secret ballot process is it takes a long time, and this gives them a lot of time to fight the union, fire the organizers, and "convince" employees to withdraw their support.

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.