[TheGreatAxios]

Closes CL-11949

The home prompt profile control stays visible when several runnable profiles exist and none is automatic. It selects the first runnable profile so a start sends that id, and the person can cycle to another. The Settings detour is not shown from this control.

Verified:

• bun --conditions=intx-src test ./apps/web/src/pages/profile-select.test.tsx — 3 pass, exit 0
• bun check --no-script -p apps/web — no type errors, exit 0
• oxlint --deny-warnings on the two changed files — 0 warnings, exit 0
• pre-commit format, lint, and typecheck — exit 0

[linear-code]

CL-11949

[corbits-triage]

Triage: Pass

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Triage will approve, request review.

[TheGreatAxios]

Screenshots of the home prompt on this branch (f900768), from this worktree's web app pointed at the existing hub. The live catalog lists two runnable profiles. Nothing between the headline and the prompt asks to choose a profile in Settings.

Model profile: Anthropic Default (control visible in the prompt toolbar):

After cycling to ChatGPT Default:

Both profiles in the toolbar list: