The problem is that if you carry the string '2021-Q1' across countries borders, the days it represents change. That makes for a very bad time interchange format.
I agree that this feature makes quarter indicator necessary. But it also makes it required that the string is something more complex.
If you carry the string '2021-10-24' across countries borders, the moments in time that it represents change as well.
That makes doesn't make it a bad time interchange format for certain use cases, though. Just like everything else with time, one must be aware of caveats. Those include timezones, leap periods, culture, DST, formats, legislation (modern and ancient), etc.
We could've gone with most of New Zealand (UTC+13 summer time) and any of the US territories in UTC-11. These are marginal cases, not exceptional ones.
I agree that this feature makes quarter indicator necessary. But it also makes it required that the string is something more complex.