[C++] Audit uses of datetime formatting

The `arrow_vendored::date` library allows us to write date/time formatting routines relatively easily, but it may throw exceptions or return silently erroneous data (see ARROW-12011). We should audit all our uses or date/time formatting routines from that library and make sure we don't let corner cases slip through.

**Reporter**: [Antoine Pitrou](https://issues.apache.org/jira/browse/ARROW-13738) / @pitrou
**Watchers**: [Rok Mihevc](https://issues.apache.org/jira/browse/ARROW-13738) / @rok
#### Related issues:
- [[C++][Python] Crashes and incorrect results when converting large integers to dates](https://github.com/apache/arrow/issues/27842) (is related to)

_{**Note**: *This issue was originally created as [ARROW-13738](https://issues.apache.org/jira/browse/ARROW-13738). Please see the [migration documentation](https://github.com/apache/arrow/issues/14542) for further details.*}