Why devotees insist that they are servants of God? Is there joy in service?

The answer is a big yes! It is because all souls serve happiness. We (souls) do anything and everything to acquire joy, happiness, thrill, peace...name it anything.

Which makes all souls servants of happiness. They work for happiness, they live for happiness. Thus everyone is servant of happiness because just as any servant works for his boss, all souls have only happiness to work for.

Happiness is God and hence it automatically follows that everyone is servant to God.

This what Hindu claims 'All the living beings and non living beings are servant to God.'

Gaurang Mahaprabhu says, 'The nature of soul is eternal servant to Krishn.'

Another view is God is the soul of soul. Because it is only through him we get all the energy that we need to do the work, to think, to see, to smell, to hear and so on.

All the the body, all the senses, mind, intellect work continuously to please 'I' which is soul. For example eyes see the objects to make 'I' happy, ear hears to keep 'I' happy. Likewise all senses do there activities to please soul. Hence all senses, mind and intellect are servant to 'I'. They don't have their own happiness.

Because God soul of soul, it but natural that soul is servant to God. There is immense happiness in service to happiness incarnate - Krishn.