Is 1/3 (1/6) such an important concept? 1/2 (1/4) is natural in metric.
I would think a bigger benefit is that you can drop up and down without any math depending on the precision you need. Or use decimals, no math involved either, since it's the same thing.
Yes, 1/3 and 1/6 are quite important. Do enough making and you will run into those fractions plenty. It just turns out that in practice, as you say, easy unit conversions outweigh the benefit of clean division. Metric's important benefit is that it's a consistent base, not that it's specifically base-10.
And now that we use computers for a lot of measurement, those infinite decimals aren't as big of a deal. (Well, most of the time, anyway. Representation of infinite fractions famously creates some programming problems. But I mean for the user making the measurement.)
Base-10 is easy to learn since it's the same we use for counting, but I think there's an interesting argument to be made that purely from a science/engineering/making perspective it would be better to use a consistent base-12 measurement system.
I would think a bigger benefit is that you can drop up and down without any math depending on the precision you need. Or use decimals, no math involved either, since it's the same thing.