Why did the Catholic Church seem to be opposed to lay people reading the Bible?

Not exactly true. Catholic Church does not opposed lay reading Bible. In fact, reading of the Bible is part of the Mass and regular attendee of the Masses will hear a large portion of both Testaments in span of few years.

Given, that Catholic rites formed in the period of nearly universal illiteracy that alone shows that Church was never against peoples learning the contents of the Bible.

However, Catholic have a different stance on interpreting it. Theologically it's based on 2 Timothy, where Paul gives eponymous Timothy right to righteously divide the word of truth and in practice it means that only the Church hierarchs (bishops, approved theologists and such) can properly interpret the Scripture and teach doctrine as spiritual successors of the Apostols.

Hence the problem Church had with the Bible translations was not laymen reading the Scripture, but laymen reading heretic interpretation of it.

Case in point- papal bull in 1713 as you name it, commonly known as Unigenitus. The bull was targeted against heresy called Jansenism, and ascetically against Pasquier Quesnel, one of the main advocates of Jansenism. Who, by the way, spread his teaching by publishing abriged version of Gospel with commentaries (Abrégé de la morale de l'Evangile).

Unigenitus condemns 101 notions of Quesnel, giving source of condemnation:

79.It is useful and necessary at every time, in every place, and for every kind of persons, to study and know the spirit, piety, and mysteries of sacred Scripture.

Pope point to 1 Cor. xiv. 6:

Now, brothers and sisters, if I come to you and speak in tongues, what good will I be to you, unless I bring you some revelation or knowledge or prophecy or word of instruction?

or the very next Quesnel proposition:

80.The reading of sacred Scripture is for all.

Pope replies with the Acts of the Apostles:

And on his way home was sitting in his chariot reading the Book of Isaiah the prophet. The Spirit told Philip, “Go to that chariot and stay near it.” Then Philip ran up to the chariot and heard the man reading Isaiah the prophet. “Do you understand what you are reading?” Philip asked.

In short, Catholic view is that reading the Bible is pointless- or even harmful- if you do not how to interpret and understand the text, and it much better leave it to the properly trained professionals something that is made even mere clear rebuffing statement 85

To interdict to Christians the reading of sacred Scripture, especially of the Gospel, is to interdict the use of light to the sons of light..

Pope responds pointing the most famous lines of Luke:

No one lights a lamp and puts it in a place where it will be hidden, or under a bowl. Instead they put it on its stand, so that those who come in may see the light.

And that essentially it: Catholic Church is not against lay people reading (or hearing) the Bible, but is very much against theologically-inept Joe's going wild with they own interpretations of it. That's the job for properly trained and educated priests.

It somewhat similar how modern scientists and academics are often opposed to amateur researchers.

P.S. In English speaking world Catholics have image of Bible haters, because of suppression of John Wycliffe, who was proclaimed martyr of reformation. However his work was suppressed not because he dared to translate Scripture to English, but because he views were, in fact, extra heresy. He even openly denied authority of the spiritual liege!