It isn't in or near South Sydney at all and its a mockery of the name. May as well just be called Rabbitohs and do away with South Sydney.
This isn't the AFL. We don't want clubs like Collingwood that play at the MCG, have their base at Melbourne Park, and have nothing to do with their district anymore. Our club names mean something. They represent the district of their name and we still have the concept of teams playing in or near the district they represent. I hate the fact that I grew up and live in the current South Sydney district, yet have to venture away from home to watch my local team play. That's why I've gone back to my original club as a kid and attend matches at Henson Park when I should be going to the SFS to watch my team in the elite league - at a venue practically in the district (less than a 400m walk from the Anzac Rd border). Even though Newtown and Souths were rivals, my kid brother never saw Newtown. He loved the rabbits though and I become a rabbitoh taking him to games at the SFS in the late 80s/early 90s every second week.
In English Football, London clubs MUST play in the borough they represent. It's why clubs such as Chelsea are finding it difficult to increase their stadium capacity, and West Ham was the preferred club to Tottenham for the London Stadium refurb. Even though they are clubs on a world-scale, they still must have this local connection to their name.