I think you have to look a little closer at your page that claims widespread acceptance. each entry is a single claim (supported only by a photo of an entry stamp page) of one person who has managed to get into that country using a World Passport. Many of the claims are obvious fakes and the photos are not of entry stamps. So for most countries, only a handful of people in the history of the passport have been successful with it. Most of those could simply be due to an immigration official not paying attention.
If you look at the Wikipedia page you will find a much more realistic assessment.
Many countries and territories have stated that they do not recognize the World Passport because it is not issued by a competent government authority, and thus does not meet the definition of a passport. As a result, travelling on a World Passport has never been easy. By 1975, Garry Davis [founder of World Service Authority] had already been imprisoned 20 times for his attempts to cross international borders with nothing but a World Passport.
The United States Department of State's official position on the World Passport is that it is just a booklet issued by a private organisation upon payment of a fee, and not a passport.As early as 1991, the US Air Transport Association specifically included the World Passport in a training film as an example of unacceptable travel papers.
In the United Kingdom, the Criminal Records Bureau of the United Kingdom's Home Office states that registered bodies should not accept the World Passport as a proof of identity, warning that "a fake ‘World Passport’ can be purchased online by members of the public and should not be confused with a genuine passport".
EDIT:If you follow the links where the WSA claims that the passports are officially accepted, many of them are letters in which the country says it accepts documents issued by the WSA as an identify document. An identity document is not a travel document, and won't necessarily get you into a country.