It did NOT until others forced the issue.
Once Scripture came widely available to the general public through translators like Luther, Wycliff, Tyndale, and Hus, most of whom were persecuted by the Catholic church for putting the Bible in the hands of the laity, THEN the Catholic church made it available to all.
The biggest claim it can honestly make is that as far as we know, it didn't outright destroy the Greek manuscripts it was in possession of.
Catholics prohibited from owning Scripture
COUNCIL OF TOULOUSE - 1229 A.D Canon 14. We prohibit also that the laity should be permitted to have the books of the Old or New Testament; unless anyone from motive of devotion should wish to have the Psalter or the Breviary for divine offices or the hours of the blessed Virgin; but we most strictly forbid their having any translation of these books.
Source: Heresy and Authority in Medieval Europe, Edited with an introduction by Edward Peters, Scolar Press, London, copyright 1980 by Edward Peters, ISBN 0-85967-621-8, pp. 194-195, citing S. R. Maitland, Facts and Documents [illustrative of the history, doctrine and rites, of the ancient Albigenses & Waldenses], London, Rivington, 1832, pp. 192-194.
The Council of Tarragona of 1234, in its second canon:
“No one may possess the books of the Old and New Testaments in the Romance language, and if anyone possesses them he must turn them over to the local bishop within eight days after promulgation of this decree, so that they may be burned lest, be he a cleric or a layman, he be suspected until he is cleared of all suspicion.” (-D. Lortsch, Historie de la Bible en France, 1910, p.14.)
As I explained before, I realize the CC was not intending to make it available. However, without them preserving the Scriptures, there would have been no Scriptures to disseminate. God used the CC in this way.