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I'm actually annoyed by this. Since the next digit is 6, he should be rounding to 3.141593, not truncating at 3.141592.
If this annoys you and so you always stop your approximations at a point where the next digit is 0..4, I think this biases your estimates of numbers (in a way that doesn't really matter for most purposes).
Why would I do that rather than simply round correctly for the number of sig figs I'm dealing with?
I don't particularly care how many digits Harry want to go to, just think he should pick 3, or 3.1, or 3.14, or 3.142, or 3.1416, or 3.14159, or 3.141593, or 3.1415927, or 3.14159265 . . . etc.
Obviously Harry should go up to the Feynman point.
It's a risk for a hypothetical person who is bothered by having to round, which isn't precisely the thing you're bothered by. A person who doesn't decide in advance how many digits to use/remember.