"Dividing it by two gives you 3/8 rather than 9.5mm." - so? What's wrong with 9.5 mm? And what's going to happen when you divide by three, not two?
"It's not like you can't do any of this with the metric system, but you'll either need to measure things more, or round things." - won't you need to round things in any positional system when you divide in not a divider of the base of the system - that is, in imperial system the base is 2, and you have convenient halfs, quarters, eights... and in metric the base is 10 and it's convenient to divide by 2, 5 and all combinations of 2s and 5s - like 50 or even 125.
In your example, 3/4 is 0.75, and divided by 2 is 0.375, and for quick estimate for what it would be in inches you take an inch as 25 mm and go from it.
Don't see anything natural in one compared to the other.