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Regarding your last paragraph, it's actually more severe:

Eventually the silent push notifications (content-available: 1) will not be delivered to the app at all. They are throttled by how much energy the app uses when launching in response.

Send too many or do too much with each one, and you'll find iOS no longer launches the app. The limits are not documented.

The maddening part is everything will work fine while you're developing and have your phone plugged in to power. You'll only notice this on battery power.