The first papyri were discovered when the villa was discovered in the mid 1700s. Attempts to unravel and read the scrolls occurred periodically for the next two centuries. While some techniques did effectively assist observers to record the words on the papyri, physically unraveling the charcoal bundles usually resulted in the papyri's irrecoverable deterioration and/or ultimate destruction. Theory of archaeological conservation changed dramatically in the later half of the 20th century, and with the electronics age a new ethos of "best to delay immediate gratification resulting in irrevocable damage when there is hope future techniques will be far superior and non-invasive" took hold and scientists began to use non-invasive techniques by the late 90s or so.*

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By the mid-2000-oughts, techniques like specialized x-ray and whatnot (the general idea across techniques has been to visually pick apart the chemical or physical signatures of ink from that of papyrus) began and new advances are made every few years.

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At present, most of the extensively studied writings seem to focus on Epicurean philosophy. There is a difference though between unraveling a scroll and destroying much of it in the process vs. trying to image and read the scroll without touching it and much of the work in the last few decades has focused on the latter. A lot of what we know of the content of the texts has come from rather destructive techniques resulting in fragmentary reconstructions of the papyri, or merely recordings of now lost material. These relatively extensive recoveries of text seem to focus almost entirely on Epicurean philosophy, both by Epicurus himself (primarily confirmed through other sources) and that of his less commonly known students. What recent techniques are doing is trying to image the surfaces of the scrolls without physically unraveling them, so that they may be preserved intact indefinitely. So far, these techniques are regularly improving, but are far from revealing easily legible sections of text. They're more proof of concept, revealing blurry letters and words inside the rolls, showing that with refined imaging and algorithms, we can one day (soon, hopefully) digitally reconstruct the scrolls as a they would appear laid out.

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Interestingly, there has been a similar debate on Egyptian sarcophagi. Many coffins for people of lesser importance in the Alexandrian/Roman period were made with papier mache of old papyri painted over with the desired imagery. Some scholars have argued that the value of the potential writing on these papyri outweighs the artistic and historical value of the intact sarcophagi and therefore they should be deconstructed. Others argue this is regressive and that future techniques would make such an approach obsolete while preserving the sarcophagi intact.

*This is part of the same trend where many archaeologists today won't excavate an entire site or feature, and many astonishing finds like the Tomb of Qin Shi Huang remain excavated - the general idea is that our inquiry and instruments will always improve, and we benefit more by being patient.

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Edit: repaired a few very tortured sentences.