Wikipedia is not an encyclopaedia
The ultimate sin on Wikipedia is original research. You know, cranks adding lengthy conspiracy diatribes to what was a perfectly reasonable article.
Banning original research is a convenient way to shut these people down. Now all you have to do is say 'where's your source?'
Unluckily for us, you can fall foul of this ban even if you do cite sources. Simply make an inference beyond what is directly stated in your sources. Usually, this looks like combining the information from multiple sources to come to a conclusion that's not in either.
If you want to include the run of a television show in an article, say, but can only find a source with start and end dates, you could easily infer the run of the show. There's really no way that your inference could be wrong here. Yet, you've still committed 'Synthesis of published material'.
In practice, people get away with doing this all the time. Wikipedia has many small articles with little oversight beyond vandalism policing. And that's a good thing. A lot more information becomes available if you can make reasonable inferences from cited sources in small articles.
The synthesis ban appeals to editors who have a quality-over-quantity view on articles. These deletionists believe Wikipedia should hold itself to the same standard as any other encyclopaedia. It's noble and it made sense in the past, when WP was conceived of as an encyclopaedia, but online.
Wikipedia's own fame and success have changed things. People are now more familiar with Wikipedia than encyclopaedias in general. Wikipedia essentially interrupted itself. Now that it occupies its own niche in people's minds, we need to rework policies to match our readers' actual understanding of what the project is.