claude-web-console: open a new session from the web UI

[studykit]

Description

Let the user start a brand-new Claude Code session from the web UI, and have it
appear in the viewer as the live session.

Context

• This capability depends on the Bidirectional input workstream (web UI →
  Claude Code control). Today the console is one-way only: hooks push
  conversation events to the browser, and there is no transport from the browser
  back to Claude Code. Bidirectional input is currently deferred and unsolved.
• Because that transport does not exist yet, this subtask is blocked until
  the Bidirectional input transport is designed and available; the approach here
  cannot be settled before then.

Acceptance Criteria

• From the viewer, the user can initiate a new session and see it start
  streaming as the live session.
• Blocked on the Bidirectional input transport — this subtask is not ready to
  implement until that dependency lands.

Related

• Depends on the Bidirectional input workstream of the claude-web-console
  epic.