We don't only depend on fragmentary ancient papyri. With only a handful of exceptions, modern editions of ancient authors are based on the mediaeval manuscript tradition -- copies of copies of copies, as you put it. Finds of fragmentary ancient copies are valuable because they act as a check on the text, but as you've noticed, they're very incomplete.

As a result there are scarcely any ancient books that survive only in ancient copies. (There are a few: Aristotle's Constitution of the Athenians; Menander's plays; the Hesiodic Catalogue of Women.) In the case of Plato's Republic, we have seven papyrus fragments found in a rubbish dump at Oxyrhynchus, in Egypt, and one other papyrus held in Milan, all dating to the 2nd and 3rd centuries CE; they're all fairly short fragments.

So the scribal manuscript tradition is where it's all happening. For Greek texts, that tradition could be either secular or monastic. You might expect 'older = better', because older means fewer stages of copying; that principle does apply to an extent, but only in a very qualified sense, because the genealogy of the manuscripts is more important -- the line of descent. A late manuscript with a high quality ancestor is going to outweigh an older manuscript with lots of mistakes. For the Republic, the oldest manuscripts date to the 9th-13th centuries; we also have parts of a Coptic translation from Nag Hammadi, and a mediaeval Hebrew retranslation of a mostly lost Arabic translation.

Here's the oldest complete copy, from the late 800s. It's held in Paris at the Bibliothèque Nationale de France, catalogued as codex Parisinus graecus 1807. The text begins at fol. 3r. At the top left you'll see the heading

+ ΠΛΑΤΩΝΟΣ +
ΠΟΛΙΤΕΙΑΙ Η ΠΕΡΙ ΔΙΚΑΙΟΥ
Α'

which is simply the title --

Plato's
Republics, or, On justice
(book) 1

And for comparison, here's an ancient copy dating to the 3rd century, corresponding to book 1, page 330 in the standard Stephanus pagination, just a few pages in from the start of the Republic. As you've noticed, it's just a fragment. But it's valuable to modern editors, because it allows them to double check the reliability of the manuscript tradition. 'Older' doesn't necessarily mean 'more reliable', but older does give extra information about the genealogy of the manuscripts, so it's always valuable.

Manuscripts aren't normally described in translated editions, because publishers of popular editions hate that kind of thing. They will be discussed in the preface to critical editions, such as the OCT edition edited by S. R. Slings, or the older edition by John Burnet. These prefaces are usually written in Latin, because tradition, so to read what Slings has to say about the manuscripts you'll need to learn Latin first. (I disapprove of this tradition, personally, because why should you need to learn two languages to read Plato? But I'm in the minority.)

I did a fairly substantial write-up offsite in 2020 about the transmission of ancient Greek and Roman literature which may help fill in some blanks.