Does the Government need to have customer service dept?

Yes it does.

Stakeholder of business organizations are limited in numbrs whereas stakeholders of a government organization are extensively large. Every individual needs to go to a government office for one work or another. Customer service department can help solve public problems in a timely manner.

The answer is definitely. Governments generally have complex processes to get paperwork done and when trying to find out out about the processes, you're more often than not given wrong information that will throw you into an endless loop. 

The duty of the customer service department would be to give clear and straight forward processes to get your request done. Also, it would be beneficial for the department to give you an idea about what papers or documents you would need to bring for whichever process you're trying to attempt finishing. 

Yes definitely.

Unfortunately, many of us have come to expect the service to be poor due to personal customer experience. All government employees needs to go on training on how to deal with people.