Ya, it can be brutal on battery.

I worked on an app once where delivery was critical, so we gave them the option of the active service+websocket, but for them the trade off was acceptable.

Pushes can be pretty flakey given all the shenanigans OEMs do on the device, even when marked as high priority correctly.

And the even worse part is when OEMs reset battery saving flags the user had set to help pushes get through and they stop working one day because of it.