`So, the Buddha goes to Kapilavatthu and stays with a former “spiritual companion” of his. The Buddha talks about teachers who focus on understanding sensuality only, feelings only, or both.
The three types of teachers as per the suttta are not as presented by your post. The sutta mentions:
1- One teacher advocates the complete understanding of sensual pleasures, but not of sights or feelings.
2- One teacher advocates the complete understanding of sensual pleasures and sights, but not of feelings.
3- One teacher advocates the complete understanding of sensual pleasures, sights, and feelings.
- Why are they arguing over what the relative should say?
From context, it is clear that Bharaṇḍu advocated that the three types of teachers have the same goals. According to the Buddha’s teachings, sensual pleasure, sights and feeling are dependently originated. As such, advocating a complete understanding of sensual pleasure and/or sensual pleasure and sights (without feelings) wont lead to liberation. I ve heard that many Hindus at the time confused Samadhi for liberation. Probably, Bharaṇḍu was one of them.
- Why doesn’t the Buddha actually tell him what the answer is and why, as he usually does?
The point of contention was whether the three types of teachers have the same goal or different goals. The Buddha provided the answer by asking Mahānāma to say they are different.
3. Why does the companion leave the city, never to return?
My own interpretation of Bharaṇḍu’s leaving Kapilavatthu is a sign of pride and shame. The context in the sutta made sure to mention his mind state/reasoning before he decided to leave:
“The Buddha has rebuked me three times in front of this illustrious Mahānāma. Why don’t I leave Kapilavatthu?”
- What is the actual answer? I’d imagine they have the same goal, though the one that teaches the understanding of both is more likely to reach it, though the Buddha seems to imply that’s not the case?
The actual answer is that the three types of teachers have different goals, not the same goal. Could you explain how they can have the same goal? Can you reach the same conclusion building on different sets of premises?