Ajax were reportedly close to roping in McCarthy to replace Dutch coach Andries Ulderink, who abruptly resigned over a disagreement with the club management on then-head of youth Hans Vonk.
Reports emerged that Efstathiou had approached the former Cape Town City mentor over coming in to lead the Urban Warriors to the 'promised land', with automatic promotion to the Premiership in sight.
However, the move did not materialise and details have since been shared as to why this is the case.
"You must realise that at the time we got the resignation from Andries Ulderink, we were sitting two points ahead with about 11 games to go – I can't remember," Efstathiou said during an online press conference with the South African Football Journalists' Association.
"So, we were a little bit concerned about what the next best move would be, because we didn't want to bring a strange coach that will try and bring different methods, and change the whole thing on a train that was running absolutely perfectly on a track.
"And we felt at the time that Benni could come in and not necessarily change what we were doing but rather bring his management skills, because I think his management and relationship with players is very good.
"So that's what attracted us to him at the time, and I did have several meetings with him and he was very keen on coming, but I think in the end of the day, the problem with his family being away and I don't think his wife wanted to return to SA, and his view that he wanted to further his career on a European stage rather than a South African stage was his reason for not coming to us.
"There was never a discussion of money; we never got to it. But I think those were the two main reasons that we didn't conclude with him. But nevertheless, it didn't happen and we gave the position to Calvin, and look at the results."