I think the intuition here is that if your error bars on the measurement are +- 40%, how can your accuracy on the error bars themselves be 4 significant figures, i.e. +-0.1%?
They aren't, but that's still the number you got. Chopping off the digits means you're slightly changing the result. It's like Tic Tac and their "0g of sugar" label, despite being 90% sugar.
No-one I've seen arguing for fewer significant digits says that a probability of 0.999 should be rounded to 1. Nor do they say it's ok to round 0.5 to 0.
If you got 10 decimal places in your calculation, it would obviously be nonsense to give all of them.