I’ve heard Amazon means “without-breast” and that Hippocrates described that they would cut off their right breast in childhood to shoot bows better. Obviously, this is just legend and doesn’t seem very practical in reality, but if the Greeks and Romans believed in it, why does it appear that no art of them depicts this detail?
Because they didn't believe it. That is: it's a myth that that's part of the myth.
Or, to unfold that a bit, it's a myth (in the sense 'popularly believed by people in the modern era, but not true') that it's part of the myth (in the sense 'a traditional ancient story about fantastic people').
It is an ancient idea. But it's a retcon -- an attempt to explain the name. The name happens to look quite like a- 'not' + mast- 'breast', that is 'breastless', and some ancient interpreters took it that that implied something about the legendary Amazons' practices. In reality that etymology is linguistically impossible, because the a in mastos, mazos 'breast' is short, but the a in Amāzōn is long, and it's a firm linguistic principle that vowels absolutely do not go randomly changing their length, ever, except in very specific circumstances.
But that didn't stop folk etymologies in antiquity. We first get the 'breastless' etymology in Hellanikos, in the 5th century BCE (FGrHist 4 F 107). He was well aware that cutting off a breast would be lethal, so he instead suggested that the Amazons used hot iron to cauterise the site of the breast before puberty.
Herodian catalogues a few alternative etymologies: (1) 'breastless', that is, lacking both breasts; (2) 'belt-wearing harvester', from ama- 'reap' + zōnē 'belt'; (3) 'with only one breast' because 'they burned one for archery'; (4) 'not eating bread', from a- 'not' + maza 'barley'; (5) 'descended from a common mother, Amazo'.
Anyway, as I say these are all retcons. They aren't part of the popular legend, though they're certainly popular among ancient interpreters of the legend. But they didn't spill over into popular sentiment, evidently, or we'd see more depictions of Amazons missing a breast. (Well, depictions of Amazons do often have one breast covered. That could be a result of the folk etymology. But it would seem the artists, too, were aware that cutting off a breast would be a good way of getting dead Amazons.)
As for the true etymology of Amāzōn, it's up in the air. There are a couple of suggestions from modern linguists. One is a group of Persian words reported by Hesychios, hamazakaran 'to make war' and mazakis 'a Persian spear' (maz- would be cognate with Greek mach- 'fight'), which suggest a possible origin in Iranian *hamazan- 'warrior'. Then again, Hesychios also reports a word hamazanides meaning a variety of 'apple tree', so you never know.
Edit: some tidying