I’m not advocating that anybody DDOS their SMS system with fake alerts. But at least an app protects against that to a greater degree than an SMS solution. The app creates a safe space for oppression.
The app is not the technology that makes this possible. It's just a webview around their website, and the SMS messages (not push notification) are sent directly to the phone via the cellular network.
Will anyone be able to silently leave the country if this app was pulled from the store? No, the SMS messages would still be sent, because they have nothing to do with the app.