This is faster :) 

No Apple Push to get in the way  

 Our push notifications get delivered the instant it hits our relay. I doubt it. It could be slightly, but likely not noticeable. 

 At least 500ms slower due to the push system pings. But much more if the phone is locked. Push is VERY slow. Especially if the device is sleeping. 

You also don't see reactions from other relays, especially DM relays that require auth.  

 Has Apple Push ever censored or banned anyone? That’s might be a better sales pitch than faster notifications. 

 if you run your own inbox relay locally, you can get notifications in literally the speed of a network packet from the sender (plus maybe a few microseconds for processing) with this, with anything else it has to go through a notification service, which means at best you get 2 packets speed (and usually more). 

If you don't run it locally, it's the time it takes the sender to send a network packet to your inbox relay (plus a few ms) + your ping to your inbox relay. 

Literally nothing can beat this, assuming perfect code naturally