Most parents who worry that Indian stories are 'too preachy' are remembering a particular kind of telling. The one where someone, at the end, says what we learned today. That telling is real. It is also avoidable.
Pick stories with movement, not morals
A jackal climbs into a vat. A monkey hangs off a branch. A boy enters a spinning wheel of soldiers. The motion is the moral. If the story moves, your child has met the lesson without anyone having to label it.
The stories that feel preachy almost always have less motion. They are made of speeches, not events. Skip them, or save them for older children who can argue with them.
Skip the post-story summary
Don't say 'and that's why we shouldn't lie.' If the story did its job, your child has felt the cost of lying, or pride, or greed, without needing to hear the label.
Children find labels uninteresting. They find consequences gripping. Trust the story. It survived three thousand years. It doesn't need a footnote from you.
Voices, not pulpits
A good narrator sounds like someone telling you a story, not someone teaching you a lesson. This is harder than it sounds, but children can distinguish the two in the first ten seconds.
If you're reading aloud, read in your own voice, not your serious-cultural-content voice. The serious voice is what makes the stories feel like homework.
Be willing to retell
A 5-year-old will ask for the same story 14 times. That repetition is them doing the moral work themselves, on their own time. You don't need to speed it up. You don't need to add new details. The 14th telling is what makes the story stick.