Spurs currently sit 14th in the Premier League table, having won just one of their last eight matches in the competition, while they were also knocked out of the EFL Cup and the FA Cup last week by Liverpool and Aston Villa respectively.
They have struggled with injuries over the last few months, with 11 senior players missing from their FA Cup squad last time out alone.
Son has been one of the players regularly available to Spurs, making 21 appearances in the Premier League with 19 of those starts.
He has netted six goals in the top-flight this term, the third-best return in the team, but has not scored since the north London derby on January 15.
Son does lead the way for assists among Spurs players, registering seven. After his assist for Pape Sarr's goal against Brentford, he became Tottenham's outright leading assister in the Premier League (69), overtaking Darren Anderton (68).
And only Dejan Kuluesevki (59) has created more chances for Spurs than him in the competition this season (37).
However, Son has been the subject of criticism for his performances during Spurs' low spell, though Postecoglou was adamant that the South Korean is the perfect player to lead the group.
"I've said a million times; we are in a world now where people are constantly wanting someone to blame. That is the world we live in," Postecoglou said.
"The reason for our poor results and consistency is not because of Sonny or my tactics.
"It is just that we have had 11 injuries for the last two months that means we have had tired players, but that doesn't suffice.
"That doesn't provide a head on a plate. That's how we live our lives these days. If someone makes a mistake, they want to tear them down. There is no leniency anymore, understanding or contextualising of why this happened.
"They are looking for someone. If people think Sonny hasn't done everything in his power to help us through this. So, what if he is a nice guy? What is wrong with being a nice person?
"Sonny is Sonny. I have no complaints about his leadership, he's handled it his way.