Require analyzer 6.5.0 and stop using deprecated nodes.

[scheglov]

• Thanks for your contribution! Please replace this text with a description of what this PR is changing or adding and why, list any relevant issues, and review the contribution guidelines below.

---

• [x] I’ve reviewed the contributor guide and applied the relevant portions to this PR.

Contribution guidelines:

- See our [contributor guide](https://github.com/dart-lang/.github/blob/main/CONTRIBUTING.md) for general expectations for PRs. - Larger or significant changes should be discussed in an issue before creating a PR. - Contributions to our repos should follow the [Dart style guide](https://dart.dev/guides/language/effective-dart) and use `dart format`. - Most changes should add an entry to the changelog and may need to [rev the pubspec package version](https://github.com/dart-lang/sdk/wiki/External-Package-Maintenance#making-a-change). - Changes to packages require [corresponding tests](https://github.com/dart-lang/.github/blob/main/CONTRIBUTING.md#Testing). Note that many Dart repos have a weekly cadence for reviewing PRs - please allow for some latency before initial review feedback.

[munificent]

You'll probably also want to update .github/workflows/test-package.yml to change 3.0.0 to 3.4.0 since we can no longer test on 3.0.0.

[scheglov]

There was two of them, I updated both.