He isn't sympathetic. He didn't savor his companionship with Killua during that whole fiasco, causing Killua to question whether they are friends. His strong rancor for Pitou estranged him from everyone that cared for him in the best way available. He doesn't know how to harness his emotions like others and valued one friendship over the other. He said he didn't need his help at that moment and he doesn't understand to killua. those are grim things to say to a person who values your company.
No Killua was the one being emo, Gon did nothing wrong. This is what happens.
Gon: "Pitou is in that building over there. Killua, lets go!"
Killua cries and thinks: "Did you mean that as a friend or as a partner?"
Like seriouslu wtf? Killua acting like an emo and desperate ex girlfriend here. Gon tells him where Pitou is and he says lets go, Killua starts crying and feeling neglected. Wtf? Seriously wtf?
You might wana mention the part where Gon says that Kite's death didn't matter as much to Killua. Damn straight it didn't! Gon only told the truth. Why must he let Pitou take care of someone precious to her? Did she think about that when she killed Kite and turned him into her puppet toy? And why did Gon have to act on Killua's words? Which were based on 'probably' and 'likely'? Gon's actions were accurate, he should have stomped her right then and there (she can't use nen wen doctor blythe is activated), drag her broken body to Peijing, and then make her heal Kite. Ofcourse the hostage plan that GON came up with was even better until Pouf spoiled it with his sneakiness. Anyway point is, at the end of the day, Gon's plan was far more accurate and feasible then Killua's 'likely' and 'probably' scenario, and Gon was right to take on Pitou alone, it was HIS revenge, not Killua's (didn't seem to give a rats ass about Kite). Ofcourse in hindsite maybe Gon should have done a little more training, coz that nen vow and restriction was crazy, or perhaps he should have set some traps like they did for Genthru. In hindsight, perhaps the whole thing could have been done differently, although honestly I think Pitou would have stomped both Gon and Killua at the same time. Gon would have ended up resorting to the same thing to win and Killua might have been dead by then.