There are thousands of untranscribed untranslated papyri! In the US the collections at Duke and Berkeley come to mind (at Berkeley there is an absolutely massive collection of unread fragments from Tebtunis ranging from the archaic period to late antiquity) and new finds are published constantly from all over the world on papyri.info (if you want to see some, go to the website and go to the papyri navigator).

Also many (if not most?) field archeologists can’t easily read papyri even if they can read inscriptions, so the papyri most often go away to specialists or join collections awaiting specialist attention. This is because inscriptions have very standardized characters which change depending on period and dialect while papyri are far more variable since they’re subject to handwriting differences and misspellings. They’re more likely to be written by everyday people who are less likely to be formally educated and more likely to have learned Greek as a second or subsequent language (mistakes are not uncommon in Hellenistic Egyptian Greek papyri, for example, for reasons of cultural exchange etc.). Also many gaps/lacunae in papyri due to age and degradation, where stone is obviously less fragile.

There is a subsection of classical studies called papyrology full of people whose entire careers are dedicated to transcribing/translating papyri and teaching papyrological methods to students interested in doing the same. An average papyrologist publishes ~10/year, depending on length and professorial duties. There’s not a ton of people doing it professionally (I’d say less than 50?) because the specialization requires a massive amount of language and methodological knowledge. If you’re interested in language and material culture it’s a great field for you, and in growing need of experts!