The Greek diaspora often had to translate its unfamiliar names into names the locals found more familiar and/or pronouncable. Hence the long line of people called Athanasios who ended up as Arthur, or Dimitrios who ended up as Jim.
Constantine was a peculiar case. As a Latin name, it should have translated into English readily, but it didn’t. There is no Western cult around St Constantine = Constantine I, so no local was called Constantine, and Constantine is a long name by English standards.
Answering a question here (Nick Nicholas’ answer to Is Kokakarsas a Greek last name?), I discovered that Constance had been used as a rendering in the 1870s in Australia. Constance is not the same name (Constantius was Constantine I’s father); but at least Constance had some usage in English in the 19th century. It fell out of fashion by the 20th century.
In Australia, the rendering of Constantine has been Con, a very Australian-like truncation of the name, which has prospered despite the fact that it doesn’t particularly hide the bearer being Greek (unlike Arthur or Jim). It’s a truncation without being a nativisation. But that strategy still requires you to be prepared to mark your name as worth preserving; I don’t believe that strategy was available in the US in the 1910s.
The choice made in the US in the 1910s was Gus. Gus is little-heard now, but it was a popular truncation of names like Angus, Gustav, or August (which were more popular then than they are now). Greeks in America grabbed on to Gus, because it was the closest they could hear to the first syllable of Kostas: /kʰɑstəs/ (in American pronunciation) ~ /ɡʌs/ (in American pronunciation), and /ɑ, ʌ/ are actually phonetically close.
This makes no sense if you have either a Greek or a Commonwealth accent of course: /kostas, ɡas/, /kʰɔstəs, ɡɐs/.
Gust Avrakotos, the CIA brains behind Charlie Wilson’s War, appears to represent the next stage of assimilation. Gustav Lascaris “Gust” Avrakotos is an odd combination of names. His father was Greek (and spare us the jokes about the surname meaning “pants-less”: it is a genuine surname in use in Lemnos). If it wasn’t his mother’s choice, I’m assuming Gustav was an elaboration of Gus, making it a distant reflection rather than a cover for Constantine.
EDIT: OP points out that Dean was available, as an English name closer to Dinos, another Greek truncation of Konstantinos (pronounced Konstandinos). It is indeed used for Constantines, and I have a second cousin in Dayton, OH called Dean. (Unlike Gus, it also is used in Australia, though nowhere near as much as Con.)
But Greeks likely weren’t thinking “I’m a Kostas, what other truncations of Konstantinos might I use that will go easier in English.” They likely stopped at “I’m a Kostas.” My impression is that the vernacular counterparts to Greek proper names were highly regionalised at the time: it just wouldn’t have occurred to the immigrants back then to switch Kostas to Dinos.