Some people I know are masters at being able to make their point indirectly, with subtlety.
Especially in a professional environment, I think this skill can go a long way. I can feel the effects of it…but still don’t fully grasp why it’s so powerful. Any insights?
WAG: Because people are more convinced of a point if they come to it themselves, or think they do, than if they are beaten over the head with it.
This is it. When someone makes a deduction for themselves rather than being told what to think, they are more likely to become invested in the belief and hold onto it.
Face-saving.
Steven Pinker talks about this in his book The Stuff of Thought, in a chapter on seduction and bribes. It’s a great book. Basically, it comes down to a matrix:
Blunt, audience agrees: they’re a little offended, but can go along with it.
Blunt, audience disagrees: they’re outraged, back against the wall, likely to “strike back.”
Subtle, audience agrees: they maintain face and go along with it.
Subtle, audience disagrees: they maintain face and put forth their own proposal without any need to “strike back.”
That’s not exactly how Pinker suggests it works, but I think it applies to this particular permutation of indirect speech.
Great insights! 
I suppose that is the same power of the ‘punch line’ in a good joke.