v0.8.63: Extract App god object state into owned submodules

[Hmbown]

Problem

crates/tui/src/tui/app.rs has a large App struct and broad impl App block. The analysis characterized this as a god object:

• roughly 150 fields on App
• thousands of lines of methods
• many concerns mixed in one impl
• a ComposerState sub-struct exists, but ownership/method boundaries are still blurry

Proposed shape

Move methods and state toward owning sub-struct/modules:

• app/mod.rs
• app/types.rs
• app/builder.rs
• app/composer.rs
• app/history.rs
• app/status.rs
• additional modules only when they remove real ownership confusion

Use an AppBuilder or equivalent for the current App::new initialization path.

Acceptance criteria

• App::new is moved into a builder/init module without behavior change.
• Composer-specific behavior moves onto ComposerState or an app/composer module.
• Status/message helpers move out of the central impl App.
• Public field/method visibility does not expand unnecessarily.
• Existing TUI app tests pass.

Risk

High. Do not start until lower-risk test extraction and renderer/API/module splits have landed.

[Hmbown]

Part of #3306.

Release-risk note: this is targeted at v0.8.63, but PRs should stay small and preserve behavior unless the issue explicitly calls out a behavior change.