That is rather incorrect. Most people in the centuries you talk about could only read a local language (where "local" could be anything from 5km² to 500km²), if they could read at all. Of course, for a long time, becoming a priest was the only way for the average person to learn to read Latin, but universities, both ones that were sponsored by the Church and ones that weren't are older than you think (14th century off the top of my head), with the Church frequently cooperating and patronising the non-affiliated universities as well.