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I'm on Verizon and mine does the same thing. Apps like eBay, PayPal, Gmail ETC are all delayed. No notifications until I actually turn on the screen. My Note 20 Ultra did the same thing.
Some apps like Ring and Aqua Mail work perfect with instant notifications but most others don't.
I've tried everything, turning off adaptive notifications, setting the the app to allow backgrounds usage ETC with no luck. Anyone have any other ideas?