How is that extra responsibility? Do you think Snapchat aren't already running servers?

And you don't care whether the time on the server is running accurately if you're just using it to generate tokens which are checked against the time on the server. It just needs to be consistent, it could be an arbitrary counter and would still work fine as long as it counted up reliably at consistent intervals.

Because instead of depending on a 3rd party, you depend on yourself (and in this case you'd hammer your own server, and notice). It is not far fetched; Windows and OS X work this way. By default, they use their own NTP servers. OS X even has separate servers for different regions (Europe, Asia, and Americas).

They could at least be good citizens and use their success to beef up the NTP pool.