[flutter_tools] Prevent crash when stdout is broken and print fails (#191356)

## Description

This PR prevents `flutter_tools` from crashing when standard
output/error is broken and the fallback logging with `print` also throws
a `FileSystemException` (e.g. Broken pipe).

Specifically:
- In `Stdio`, introduced `_safePrint` which catches and ignores
exceptions thrown by `print` when stdout/error streams are broken or
closed.
- Replaced fallback `print` invocations in `stdoutWrite`, `stderrWrite`,
and `_stdioWrite` with `_safePrint`.
- Added unit tests in
`packages/flutter_tools/test/general.shard/base/io_test.dart` verifying
that `stdoutWrite` does not crash when writes fail asynchronously or
when the stream is already closed.

## Related Issues

* Fixes https://github.com/flutter/flutter/issues/51872

## Pre-launch Checklist

- [x] I read the [Contributor Guide] and followed the process outlined
there for submitting PRs.
- [x] I read the [AI contribution guidelines] and understand my
responsibilities, or I am not using AI tools.
- [x] I read the [Tree Hygiene] wiki page, which explains my
responsibilities.
- [x] I read and followed the [Flutter Style Guide], including [Features
we expect every widget to implement].
- [x] I signed the [CLA].
- [x] I listed at least one issue that this PR fixes in the description
above.
- [x] I updated/added relevant documentation (doc comments with `///`).
- [x] I added new tests to check the change I am making, or this PR is
[test-exempt].
- [x] I followed the [breaking change policy] and added [Data Driven
Fixes] where supported.
- [x] All existing and new tests are passing.

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[Tree Hygiene]:
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[test-exempt]:
https://github.com/flutter/flutter/blob/main/docs/contributing/Tree-hygiene.md#tests
[Flutter Style Guide]:
https://github.com/flutter/flutter/blob/main/docs/contributing/Style-guide-for-Flutter-repo.md
[Features we expect every widget to implement]:
https://github.com/flutter/flutter/blob/main/docs/contributing/Style-guide-for-Flutter-repo.md#features-we-expect-every-widget-to-implement
[CLA]: https://cla.developers.google.com/
[flutter/tests]: https://github.com/flutter/tests
[breaking change policy]:
https://github.com/flutter/flutter/blob/main/docs/contributing/Tree-hygiene.md#handling-breaking-changes
[Discord]:
https://github.com/flutter/flutter/blob/main/docs/contributing/Chat.md
[Data Driven Fixes]:
https://github.com/flutter/flutter/blob/main/docs/contributing/Data-driven-Fixes.md