Migrate remaining eligible private members to ECMAScript syntax

[Bharat Middha (bmiddha)]

Summary

• migrate the remaining eligible TypeScript-private fields and methods reported by @rushstack/prefer-ecmascript-private-members to ECMAScript #private syntax
• update internal references and replace test/mock access to private implementation details with behavior assertions or explicit internal test seams
• preserve ES5 output in the affected legacy web samples by using protected handlers instead of native private identifiers
• include the ESLint 9 fixture migration currently carried by Require ECMAScript private class members #5945
• add Rush change descriptions for affected published packages

This is a prerequisite for #5945.

Validation

• targeted builds passed for all affected Rush projects
• targeted tests passed for @microsoft/rush-lib, @rushstack/rush-reporter, @rushstack/rush-terminal-renderer, rushd-wire-e2e-test, @rushstack/rush-amazon-s3-build-cache-plugin, @rushstack/rush-buildxl-graph-plugin, and eslint-9-test
• rush change --verify passed
• repository-wide rush build passed after temporarily overlaying only Require ECMAScript private class members #5945's ESLint rule/config changes, with 0 @rushstack/prefer-ecmascript-private-members warnings

Intentional exclusions

• libraries/node-core-library remains unchanged because its declarations must remain compatible with ES5 consumers
• ES5-targeted Storybook v6, Heft web-rig, and webpack tutorial handlers retain TypeScript-emittable syntax via protected members rather than changing their compilation target