You're missing it entirely. If we weren't measuring using meters in the first place he wouldn't even be asked to cut 1/3rd of a meter. He'd be asked to cut 1/3rd of a yard, which is very easy on a yardstick. It only needs to fit in a specific spot that is a third of a meter because the structure around that spot was measured and built with the metric system.

But they wouldn't ask for a third of a meter if they were using metric, just like you wouldn't ask for a firth of a yard if you were using imperial.

Yes they would, if they wanted something divided by 3.

The US customary system conventionally uses reciprocal powers of two. 1/3 is unlikely to be a specified distance. I rarely hear anyone speak in yards. If someone wanted “a third of a yard” they’d just say one foot. A third of a foot is just four inches. Or slightly less for clearance would be three and 7/8 inches, etc.