Is it possible to make a New York "style" pizza using a pizza disk or screen? You betcha! You could probably make it on a block of wood too. The operative word being "STYLE". To a purist, the answer is emphatically NO! but we are not talking "New York" pizza (authentic), we are talking New York style. Just what is a New York style pizza??? There in lies the issue. The word style is open to interpretation. Just a couple of meanings are: In the manner of, or in the appearance/likeness of.
Tom, we're not lawyers in a courtroom stretching the meanings of words as far as they can possibly be stretched in an effort to find legal loopholes. If it could help them win their case, a lawyer could show you a thousand exhibits why 2+2=5. For those that are the least bit familiar with pizza, 'NY style' pizza = NY pizza. The term 'style' was introduced not as a means of producing a looser definition (as it's occasionally used by lawyers for some commercial foods), but for geographical purposes. If you're making the exact same pizza that they make in NY, but you're in, say Des Moines, you wouldn't advertise 'NY pizza,' but rather 'NY style pizza.'
'NY style' pizza = NY pizza = authentic NY style pizza
'Neapolitan style' pizza = Neapolitan pizza = authentic Neapolitan style pizza
If you start playing lawyer games with the term 'style,' then the manner in which these sacred foods are defined, foods that millions of people have helped to define, becomes meaningless. Once you say that 'NY style' can basically mean anything, it's the tower of Babel, it's confounding speech, it's chaos- and you can kiss this cultural treasure goodbye.