This doesn't really answer your question exactly but I would imagine it had a lot to do with the lack of a major industry creating and dispersing writing implements, primarily inks. I wouldn't attribute it to a social or cultural aspect of the Greek Empire that emphasized being able to read without the subsequent knowledge of how to write. Just think, if you were a perfectly capable reader but had never written before, don't you think you would still quite easily be able to replicate the symbols to form words you already know reflexively? A simple answer seems apparent: educators wished to enlighten younger generations but did not see a great benefit from their writings that would outweigh the costs of widespread diffusion of writing materials. Creating written works was once an incredibly time intensive task, so it was more of an art form than it is now. By that I don't mean the creation of the content of the text but the actual physical form of the text, which is why we see so many instances of monks and others spending a large portion of their time methodically replicating a book that they already have.

If I want my child to learn to write, all I have to do is supply him with very cheap materials available at basically any store that sells general goods. For earlier man it was much more difficult, and much more like a craft than it is today where machines churn out thousands of pages of paper a minute. Inks would have similarly been difficult to acquire, and although there were a huge number of sources for inks (such as plant dyes, ore/minerals, and even animal sources) it is still quite apparent that the average person isn't going to give much of a fig about making uniformly colored bottles of some liquid so that a child can learn to properly write. Obviously affluent and well-connected individuals would be the exceptions to this rule, as many of them would enter either the secular or regular clergy (at least in the Latin West) and most likely learn to write there at some point.

Some of this is clearly conjecture, and I apologize for that. The only source I have that mentions anything like this is A World Lit Only by Fire by William Manchester.

I'm just speculating here.

I think, by its nature, writing is useful as a one-to-many communication medium, meaning that one person could create signs, or a list, or a letter, and then many other people could read each written item. This scales, and reading skill becomes useful more quickly, easily, and widely than writing skill does.