Not entirely sure about the de-google’d version of the Home Assistant companion app, but I know the regular companion app uses Firebase (and whatever the Apple equivalent is called, I forget) to deliver notifications, and it still would using Telegram as Telegram also uses Firebase. Apprise is a bit different as it can use multiple backends. Regardless, there are multiple ways to do things. Ntfy iphone and google app do not route your data through a third party server. I self host the ntfy server on my own machine and my phone connects to it and receives data. It also provides a nice UI akin to Pushbullet I can use to send myself stuff privately.

You can’t replicate all of what ntfy does with Home Assistant. There’s more to it than just delivering notifications, it’s the whole app frontend and persistent data etc. If it’s not clear to you what it’s for from my description that’s fine, maybe go look around at it yourself.