Hey there Atahan,
It's not that the google app is running in the background forever, its just that the google app is a native experience whereas the Kagi app currently has to load a webview and such before showing you the search bar.
We do have plans to make our app native as well, just like the google app so you can instantly start searching without waiting.
I can't give a timeline on that currently but I just wanted to note that it is planned 🙂