That article is a bit weird (who questions the existence of the Library? And what exactly does it have to do with this mummy--the idea that Aeschylus' plays were lost when the Library burned is baloney. And what does this guy mean when he says a "faithful adaptation?" Faithful to what exactly? Oh, and for the record there's no such play as the Achilles, the trilogy dealing with Achilles was the Achilleis and contained the Myrmidons, Nereids, and Phrygians--which play is it on these fragments?) but this is hardly unusual for Egyptian papyri to turn up as debris thrown away. Papyrus was plentiful in Hellenistic and Roman Egypt, so plentiful that it was often thrown away or torn into scraps and strips to be used for other purposes, especially packing. Our largest source of papyri is at Oxyrhynchus, at a literal garbage heap. We have millions of papyrus fragments, the vast majority of which are little more than a few chewed-up scraps of some unimportant local administrative document or somebody's will--interesting for some scholars, but often not worth the time, effort, and money it takes for a papyrologist to go through all of that to look for something that nobody but a few is interested in. Mummies are a surprisingly common source of papyrus fragments, as mummies were often stuffed (not, to my knowledge, wrapped as the article suggests--wrapping was done with linen usually, and given the other weirdness in the article I think the author has to be mistaken) with papyrus strips--it was around and it made a good packing material.

Since you mention the possibility of discovering documents, I should say that it's highly unlikely in the extreme that we find major texts complete or nearly complete from papyrus fragments--at least, not regularly. Papyrus fragments are generally pretty...well, fragmentary, often preserving little more than a few words. In cases of larger fragments what we get is not always even. In Egyptian papyri you're as likely to find a bit of a letter as you are to find a scrap of a tragedian--exciting for some (e.g. for me), but given that papyrology is extremely exhausting and consumes resources that could be going elsewhere it sort of puts a bottleneck on what we can analyze. But the big thing here--and I should stress this--is that our complete texts, with few exceptions, are not found on ancient manuscripts. Nearly every text that a classicist studies is known to us through medieval manuscripts, usually no earlier than like the 8th or 9th Centuries, generally through multiple copies that are then compiled and compared for errors. Papyrus is pretty important (especially for spotting possible errors in later texts) but it rarely provides us with complete texts, and when it does (for example, the recent works of Sappho that were identified) it's generally fairly small texts, and even then with lacunae and other issues. It's certainly possible to find major works nearly intact from papyrus--Menander's Dyskolos is known only through a nearly complete papyrus from the Bodmer Papyri--but it's uncommon, and in the case of the Dyskolos and other large fragments it's rarely because the papyrus was in a trash heap or was shredded up for packing material. Not surprisingly, shreds of papyrus don't tend to yield much that's useful, or even readable.

Still, the possibilities of papyrology are very exciting. We have no idea what we might dig up or where, or what might be lurking in a papyrus collection at some university. Complete, or even lengthy, finds are pretty rare though, and have never made up more than a tiny minority of the manuscripts from which classical texts are compiled. More recent attempts to analyze genuinely ancient manuscripts that we have have also been promising. I think it was last year that some guys at Naples were able to use X-rays to make out some of the text of some of the Herculaneum scrolls (which are not really papyri anymore, they're more like bricks now), which could be very fruitful if it's perfected (and Richard Janko seems confident, and I usually trust him)--for decades there's been tons of speculation about what might be in those papyri besides the Epicurean fragments that have made up pretty much everything we've been able to read of it.

Mummies would frequently have prayers and info inscribed on various components, such as the sarcophagous. The Book of the Dead was also a common burial artifact.

As for the details, it all depended on the weath of the deceased. Because mummification was done on all classes, weathier people got the "deluxe plan." Others would often get a cheap version. This would include both the quality of the wrappings, and also the steps taken during preservation, such as materials used, and having scribes write prayers to be included.

Wrappings that included greek writings would clearly have been considered cheaper at the time. That's a recycle job, which was common.

I do think it is a terrific way to find written material from the past. I am just not sure that I would want the mummies damaged. They would need to be scanned somehow.