DeltaChat can only move the messages in the deltachat folder when it is running.
If you don’t want any messages in you inbox even when deltachat is not always active, you need to create a filter rule on gmail that moves the messages. https://support.google.com/mail/answer/6579?hl=en
The filter needs to say:
If Chat-Version header exists move the email to the DeltaChat folder.
(I don’t know the exact steps as I don’t use gmail, it could be that gmail doesn’t support filtering emails based on headers in that case you would need to use another email provider if you absolutely need that behaviour)