Benfords law only applies to certain sets of numbers.
Not all numbers display Benfords law. Therefore Benfords law should not be considered when talking about rounding unless there's a reason to think that it applies to your particular set of numbers.
It would be interesting to know what sets of numbers do not follow Benford's law.
Check Knuth's discussion of it in "The Art of Computer Programming, section 4.2.4 where he talks about the distribution of floating point numbers. Benford's research took numbers from 20299 different sources. Additionally, the phenomenon is bolstered by noticing, in the pre-calculator days, that the front pages of logarithm books tended to be more worn than the back pages.
The whole discussion covers about four pages, and contains what I think of adequate mathematics demonstrating the point.
The interesting thing about Benford's law is how widely applicable it is. Most sets of numbers you come across in your life are likely to follow it, or at least have leading 1s be more common than leading 9s. As such we should definitely be considering it when trying to decide which rounding rule to adopt in the general case.
I'll try to make the parent's point a different way. This is about rounding. That's about the digits with the least significance. A good rounding scheme wants to see a uniform distribution there in the two least significant digits. Benford's law breaks down after four digits pretty much completely. That's the over-thinking it aspect.
Benfords law only applies to certain sets of numbers.
Not all numbers display Benfords law. Therefore Benfords law should not be considered when talking about rounding unless there's a reason to think that it applies to your particular set of numbers.