I mostly teach physics, not math, so I get a lot of experience in my lab classes where I see students manipulating numerical data. In particular, I often see them do numerical calculations that are long enough so that rounding errors really could accumulate. (I doubt that math teachers very often see their students do a really lengthy numerical calculation.)
The thing that stands out to me most from this experience is that most students at this level (freshman college STEM majors) show an almost complete inability to reason about how a calculation works. This extends to the most incredibly simple concepts, such as the idea that when you change the input to a calculation, the output will also change.
As an example, I will often see students write down intermediate steps in their lab notebooks, recording all the digits from their calculators. As an imaginary example, suppose that they've just immersed a cube in water and measured its volume to be 124 cubic centimeters. They next use this to calculate the length of the cube's edge, which is going to be a step in a longer calculation. They write it down as 4.98663095223864 cm. When I ask them why they're doing that, here are some typical answers I get:
Answers like these show that they're not reasoning at all about how the calculation works. They expect to be told an arbitrary rule to follow, which will ensure them a good grade.
For these reasons, I agree very much with Sue VanHattum. When something like this comes up, its value lies in the fact that we can use it to guide students toward thinking more deeply about what would make sense. Teaching the student a more statistically desirable procedure should not be the point, unless the student is ready to make sense of the procedure. Sometimes we can say things to our students that will nudge them into thinking:
When I was a kid, my family got our first electronic calculator, and it had a 6-digit display. I see that you iPhone calculated that cube root with 14 digits of precision, and you wrote them all down in your lab notebook. If you got a newer phone and it displayed 100 digits, would you write all of those down?
This other lab group measured 123 cm$^3$ for the volume of the cube. If we took the cube root of that number instead of your 124 cm$^3$, which of those digits in your result do you think would change?
As a practical matter, for people who are more numerate, I don't think it makes a lot of sense to follow any particular rule for rounding fives. In almost all real-world applications, the answer is that if rounding a 5 up or down is enough to make a difference in your life, then you're rounding too much; you should retain more digits of precision.