Android could perfectly well let you give an app local permissions without giving it call-out-to-the-network permissions. Snapsave shouldn't need to ever call out to external servers in the first place, if it does only what it advertises.
Android doesn't do this because of their broken ad-based ecosystem, though: they don't want to draw your attention to apps that unnecessarily call out to the network, because the most common reason for doing so is to show ads.
It's true that without controlling the endpoints, Snapchat can't stop one particular attack vector: the people who control those devices saving images themselves. The usual "DRM" problem.
But what seems to have happened here is that users installed an app which, unbeknownst to them, sent copies of the images to a third-party server. That threat model is possible to guard against, although it's arguably more an issue with Android than Snapchat that something like that easily happens without users noticing, because Android's app-permission model leaks like a sieve.
It's used in a ton of places in the bowels of big companies as well.
Apple does that too, though on end-user machines. When connecting to wifi, it doesn't enable the connection until it first verifies you're really connected. It does that by trying to pull a specific known Apple URL. If it doesn't get the expected contents, it guesses you're behind a wifi hotspot's login wall, and pops up the "please log in" page. The intent of this is to make sure apps like Dropbox and your email and whatever don't think they're back online and start failing connections, in the time between when you connect to a hotspot wifi and when you log in. But it also means that if Apple's URL goes down, wifi connection will end up with extra hoops to jump through to get it to work.
I don't have any experience with the military, but I do have experience working with defense contractors on DARPA projects, and in that context I have not been very impressed.
calamari is squid, not octopus!
If A = B and B = C, then A = C, except where void or prohibited by law. -- Roy Santoro