Even if the system app doesn't utilize much of it's permissions, it still at least won't be killed or denied from using microphone just because it's in the background or because the phone display is turned off.

And again - how do you achieve Google Assistant-like direct voice activation without system permissions? How do you get rid of Google Assistant without having a ROM that doesn't include it?

Yes perhaps building a standard AOSP and adding their app is enough, so you could flash your phone with a simple custom ROM to achieve what they did. But that is still significantly different from just installing the app on a standard Android phone.

BTW stuff like Google Assistant, Play store etc is not part of BSP or AOSP. The standard apps they refer to are things like the basic browser (not Chrome), dialer, launcher, settings and so on.

Also, it's possible their app wouldn't get through Play store review due to privacy/security concerns.