Christianity is not a "go it alone" sort of religion. We are meant to be in community and work with community. So, I feel like biblical illiteracy is a matter of a failure on the church's part to properly discipleship people.
There are some churches that have a poor curriculum for their youth. It's possible for faithfully attending families to get their children biblically literate through a good program that surveys the scriptures over the course of several years.
For people who are unchurched and convert as adults, there should be adult bible study, small group, etc. available that surveys the bible every year.
Also, there are many churches that follow a calendar for Sunday readings. In the Anglican church, for example, if you faithfully attend each Sunday for three years, you will hear the entire bible read (now, you won't necessarily be studying each part of the bible, but you'll have heard it all). It doesn't seem like it would be that hard for a church to make a commitment to working its way through the entire bible in the course of three years (and by "church" I mean that the pastor would follow a calendar and preach on a little bit each week, so the congregation can have readings and teaching together).
There are lots of reasons why individuals might not dig into the bible on their own. Lots of people are intimidated by it. Maybe they aren't big readers and it seems difficult to understand. They need other believers to come along side them and help them gain a love for the scriptures.