Would a king regularly be a shepherd who tends his flocks?  Especially one who is the richest man in the world such as Solomon?

Shepherd?

The man in this poem whom she loves is certainly a shepherd:

• Song of Songs 1:7
  Tell me, you whom I love, where you graze your flock and where you rest your sheep at midday. Why should I be like a veiled woman beside the flocks of your friends?

And he isn’t one who just visits his flocks as a king, but one who is with them, caring for them, causing them to rest.  That and she has to go look for him among the streets at one point.  Wouldn’t the guards know where a king was at?

King?

And then there’s the king imagery.