[image_picker] Adopt readme excerpts

[stuartmorgan-g]

Converts the README from manually-maintained code snippets to using excerpts from a code file.

Includes sanity check tests of the snippet code to make sure it runs successfully.

Part of https://github.com/flutter/flutter/issues/102679

Pre-launch Checklist

• [x] I read the Contributor Guide and followed the process outlined there for submitting PRs.
• [x] I read the Tree Hygiene wiki page, which explains my responsibilities.
• [x] I read and followed the relevant style guides and ran the auto-formatter. (Unlike the flutter/flutter repo, the flutter/packages repo does use dart format.)
• [x] I signed the CLA.
• [x] The title of the PR starts with the name of the package surrounded by square brackets, e.g. [shared_preferences]
• [x] I listed at least one issue that this PR fixes in the description above.
• [x] I updated pubspec.yaml with an appropriate new version according to the pub versioning philosophy, or this PR is exempt from version changes.
• [x] I updated CHANGELOG.md to add a description of the change, following repository CHANGELOG style.
• [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] All existing and new tests are passing.