refactor: minimize suggestion menu host markup

[bartekplus]

Summary

• minimize the shadow-path suggestion popup host to an anonymous div with only a generated id
• move early-Tab visibility checks from host attributes onto input-side state
• update unit and e2e helpers to resolve menu hosts from managed inputs, including shadow-root cases

Test Plan

• bun test tests/SuggestionMenuView.test.ts tests/SuggestionMenuPresenter.test.ts tests/EarlyTabAcceptMainWorldBridge.test.ts tests/SuggestionManagerRuntime.test.ts tests/SuggestionManager.test.ts
• bun run test:e2e
• bun run check (blocked by pre-existing TypeScript errors in ManualAttachUiManager, SuggestionManagerRuntime, and SuggestionTextEditService)