refactor(tui): centralize blocking interactions

[harryalbert]

Description

Linked Issue

• The linked issue is labeled ready-to-spec or ready-to-implement.
• Where appropriate, screenshots or a short video of the implementation are included below (especially for user-visible or UI changes).

Testing

• I have manually tested my changes locally with ./script/run

Screenshots / Videos

Agent Mode

• Warp Agent Mode - This PR was created via Warp's AI Agent Mode

[harryalbert]

• feat(tui): add local-to-cloud handoff #14208
• refactor: share local-to-cloud handoff pipeline #14207
• refactor(tui): centralize blocking interactions #14206 👈 (View in Graphite)
• master

This stack of pull requests is managed by Graphite. Learn more about stacking.