The main reason for GySEVs peculiar existence is that the current Austrian Land of Burgenland was historically part of the kingdom of Hungary - in the last decades of Austria-Hungary railways were a „devolved matter“. So originally, it was a Hungarian railway which just crossed a couple of kilometers into Austria at Ebenfurth.
In 1921, Burgenland (minus some important parts, mainly Sopron - historically Ödenburg in German, even though that name is rarely used anymore) came to Austria and GySEV was really bi-national, since of course Austria and Hungary where by then separate countries.
After WW2, GySEV remained a public limited corporation because their Austrian concession was based on this status. However, with collectivization, all shares were owned by the Hungarian state. Austria only acquired part of the shares after the fall of the Iron Curtain.
Sopron was the main town in Burgenland when it was still Hungarian, so it was the railway hub as well. So no way around it when you wanted to go to Deutschkreutz. Indeed, that line, after 1921, did not (and still does not) have a direct connection to the rest of the Austrian network except through Hungary (towards the south, the line used to go on to Szombathely, but this was closed long ago).
Indeed for this reason the short stretch from the border into Deutschkreutz is electrified with 25 kV. The rest of the line was, unfortunately but not surprisingly (the detour via Sopron makes journey times unattractive) closed in stages long ago.
Very interesting, how the GySEV anomaly thus originated -- thank you. I do get the picture that the "dual" aspect of Austria / Hungary up to the empire's dissolution post-World War I had various strangely complex aspects; with, shall we say, assorted sources of controversy / discontent both "before" and "after", concerning "where was whose".