So,shouldn't battery generate an electric field directing from the positive side to the negative side?
That is indeed the case within the battery.
An electrochemical reaction within the battery moves charges within the battery so that one terminal has a deficiency of electrons (the positive terminal) and the other terminal has a surplus of electrons (the negative terminal).
That migration of charges produced an electric field within the battery from the positive to the negative terminal.
This electric field opposes the migration of the charges within the battery and, if the battery terminals are not connected externally, stops the migration so a steady potential difference is set up between the terminals of the battery.
When a conducting circuit is connected to the battery terminals then charges move externally and the electrochemical process within the battery tries to maintain the potential difference across the terminals of the battery by making charges migrate within the battery.