Short version: the Reformation happened.

Luther wasn't the first small-r reformer of the Middle Ages and Early Modern era to advocate for accessibility of scripture in local languages other than Latin -- Jan Hus advocated for individual study of scripture for lay people, including women, in medieval Bohemia, and John Wycliffe before him undermined the unique privileges of clergy including their exclusive access to scripture. Scripture wasn't completely inaccessible to all lay people prior to this in such a way that integral elements of the life of Christ, for instance, would necessarily blindside a first-time reader. Lay people would have been familiar with Bible stories not only from religious images and sermons delivered by priests but also religious books for lay people and in some regions religious theater -- rather than just hearing your priest describe the story of Abraham and Isaac, you can also see the story of Abraham and Isaac performed by a local guild in a nice mystery play, in a relatively orthodox interpretation of those events. (Though the timeframe in which mystery plays are documented in writing and the timeframe of the Reformation do overlap.) This still isn't the same as freely accessing scriptural texts themselves in the first place, in one's own home and at one's own leisure, which would allow for more fine-grained theological analysis regardless of language even for people without formal theological education.

Part of the problem was not simply the language issue but as you say, scripture being distributed broadly for the first time as a mass-produced text and not in hand-written volumes earmarked for monastic use. But an approximate familiarity with informal Christian narratives and storytelling did lessen the chances of a first-time scripture reader being blindsided by what's in the Bible. A first-time scripture reader might, however, be struck by what's not in the Bible -- aspects of Catholic interpretation that are absent or weak from plain scripture alone without the theological scaffolding of additional approved interpretation. No direct references to indulgences or to purgatory, for instance, and the scriptural foundation of the doctrine of transubstantiation considerably more doubtful than Catholic tradition would have it -- these were not minor issues in the late Medieval/(very) Early Modern Catholic Church, and a moderately educated reader being able to access the texts of scripture stripped bare of explicitly Catholic and papally-approved glosses paved the road to multiplying scriptural interpretations. If you're a traditional Roman Catholic facing down multiple Protestant sects in this era, that's a seriously bad thing -- those new interpretations are very likely heresy, and it's critically important that such thinking be brought in line with official Pope-approved doctrines or there will be grave consequences. If you're a Protestant, in addition to despising your own fair share of heresies you are likely now more vigorously equipped than ever to scrutinize the decisions made by nominally Christian organizations (like your local church institutions, or the Church at large) and how they measure up to the precedents of the Old and New Testament. Furthermore, the combination of the Bible translated into local vernacular languages and the rise of the printing press made it much easier to propagate choice passages of scripture to reinforce a writer's points, and much easier for these commentaries (often in pamphlet form) to be read by people who did not or could not own their own Latin-language Bible.

Of course, none of that is any good if people aren't literate in the vernacular language of their region -- if you can read neither German nor Latin, a German-language volume of the New Testament won't do you a lick of good. But as general literacy in vernacular languages as well as Latin increased, so did demand. Availability of scripture in the vernacular wasn't the sole cause of the Reformation, but it sure did grease the wheels.

Some sources:

• A Blessed Shore: England and Bohemia from Chaucer to Shakespeare by Alfred Thomas (its chapters on Jan Hus illustrate how well scripture in the vernacular, not only for non-clergy but for women, went over among the Hussites' contemporaries -- there's an incredibly inventively nasty poem that basically accuses Hussite women of only saying they want to study the Bible in the vernacular so they can get inappropriate power over men, including one woman inviting a guy back to her private quarters for a Bible study, wink-wink, and offering to let him see her nice creamy pair of Testaments)

• If you want a massive overview of the various religious and political shifts making up the Protestant Reformation, including aspects directly related to the new accessibility of vernacular translations of scripture and Catholic responses to that, Carlos Eire's Reformations: The Early Modern World, 1450-1650 is really interesting (though I have my beefs with it) -- Bruce Shelley's Church History In Plain Language is a pretty lucid and much briefer overview. Pretty much any overview of the Reformation has to touch on this, that's how impactful this new accessibility was.