Scheduled tasks (routines): show and allow choosing the model per routine

Type: Feature request

Area: Scheduled tasks (routines) in the Claude desktop app / Claude Code (scheduled-tasks MCP server).

Problem

There's currently no way to see or set the model a scheduled task (routine) uses:

• list_scheduled_tasks (MCP) returns schedule, cron, enabled, lastRunAt… but no model field.
• update_scheduled_task (MCP) has no model parameter → the model can't be changed via the API.
• In the app, the model actually used only shows up inside each run's session config JSON (AppData/Roaming/Claude/claude-code-sessions/.../<session>.json, model field). Changing the model on a session (or a fork) does not reliably repilot future scheduled runs.

Result: users can't audit or control the per-routine model choice. With 20+ routines this matters for cost (Haiku vs Sonnet vs Opus) and quality.

Request

1. Display each scheduled task's model — in the routines list / task detail, and in list_scheduled_tasks output.
2. Set / update it — a model field on update_scheduled_task, and a model selector in the routine settings that persists to future runs.

Why it matters

• Cost tuning: route simple digests to Haiku, reasoning-heavy routines to Sonnet/Opus.
• Easy migration when a new model ships (e.g. Sonnet 4.6 → Sonnet 5): today there's no reliable way to switch a routine's model.

Related: #64843 (also area:routines — scheduled task sessions not shown under the Scheduled sidebar section).