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How? Especially the less battery part. How? Will i miss out on live notifications?

You get an option to use SMS for notifications

Does that seriously safe battery? I would have thought push notifications are the most optimized thing ever.

SMS is one of the options(If you are in power savings mode or switched off your mobile data because if low battery)

Less battery is used because there is no active service in background that keeps connected to server.Like WhatsApp or FB does.

Hence using less data and less battery