I found the original article a bit strange in it's partitioning of the world into { US, Europe, some places }, I feel that is giving ... undue weight to the US convention, but I haven't studied this to see if I'm right.
I question whether "Thursday of week 37" is really that much clearer than "Thursday, September 13".
It never occurred to me to check my Apple Watch for a week number complication. A bit disappointed it's not there, even if I'd never use it.
Update: you can install an app to Apple Watch that supplies a complication for it. "Current Week".
And sometimes you really want to speak about the entire week without naming a day ("we'll have the first boards week 40, smoke-test and bring-up during week 41, and then start proper development by week 42"), since it's times in the future and you don't want to be too specific but still align work tasks with weeks.
Also (re: a sibling comment) Google Calendar shows week numbers, or can be configured to. For quickly finding out the week number, there's always https://vecka.nu. :)
KDE and i3 desktops can be configured for pretty much arbitrary date formats, I always have the KW in there.
For instance, if I were to tell you "Beginning of next week, I'm gonna X", you'd understand Monday, correct?
I'd likely understand it as Monday in other contexts too, because it's very rare for something to start on Sunday. Monday and Saturday are the transition points and thus the logical places to start doing something.
But I might understand it as Sunday if context seemed to point that way. What I'm saying is that nothing is organized according to "the beginning of the week", so the question of which day is the beginning of the week doesn't have any implications.
Why would this be unusual? I'd expect pay-by-the-week to be more common for part-time employees, but it's not rare for FTEs either.