Arresting him quietly without the media gets them Snowden, but not the deterrent.
Whistle-blowers may be releasing information because of strong ethical objections, not rational thoughts about their own well-being.
I think watching this unfold with obvious lies from governments has actually strengthened the ethics in releasing this information, and may reinforce others with similar information that leaking really is the morally right thing to do.
Same with Snowden - creating a diplomatic incident that now has pretty much every Latin American leader screaming bloody murder and have officials describing it as virtually a kidnapping, and complaining about colonialism, means European countries and others will quickly start seeing that there's a very high potential price to pay for bending to US demands in this, in a case where they likely have little sympathy to the US position in the first place given that pretty much all of them have faced revelations of US spying targeting them. If the US keeps pushing like that, they might just find that someone will snap and decide to tell them to go to hell.
It'll be interesting to see if they manage the balancing act of pushing just hard enough to keep everyone in line without pushing anyone too far.