Header CTA cleanup: remove Deploy CTA, add Deployments tab + status badges + overflow menu + Publishing UX

Please select the area the issue is related to

AI Workspace

Please select the aspect the issue is related to

Aspect/UI (Frontend layouts, components, styling)

Description

Remove the redundant/legacy header CTAs ("Deploy to Gateway", standalone Delete) from LLM Provider and App LLM Proxy overview pages, and replace with a coherent tab + badge model: a dedicated Deployments tab (relocating the existing deployment card/flow), status badges (deployment + publish state) in the header, and an overflow (...) menu for secondary actions.

These are tightly coupled and should ship together as one release. Removing the header CTA before the tab/badges/overflow menu exist would strand users' access to deploy/delete functionality, and shipping Publishing UX separately afterward would mean a second round of header churn (see linked Opens on Publishing Status / Tabs).

Code grounding: Deploy CTA exists on LLMProxyOverview.tsx and ServiceProviderOverview.tsx only (not Gen AI App). No overflow menu exists anywhere yet. Deployment info currently lives in a card inside the Overview tab (LLMProxyDeploymentsCard.tsx / ServiceProviderDeploymentsCard.tsx) that can be relocated rather than rebuilt. A deployment-status chip pattern already exists for Providers but is currently dead/commented-out code in ProvidersList.tsx.

Product Management Issue: none yet, to be added.
Feature Proposal Document: AI Workspace IA (Notion), https://app.notion.com/p/3a4bc54205f7808194b5c3175f91e3a7

Sub-tasks:

• Remove "Deploy to Gateway" header CTA (LLMProxyOverview.tsx, ServiceProviderOverview.tsx)
• Add "Deployments" tab, relocate existing deployment card content — Figma: already drafted
• Add deployment-status badge to header + list rows (reuse/un-comment existing statusChipColor pattern in ProvidersList.tsx; extend to Proxies/Apps) — Figma reference
• Add header overflow (...) menu; move Delete (and future Unpublish) into it — already drafted: Provider, Proxy
• Design + ship Publishing UX (publish/unpublish flow, publish-status badge) as part of this same release — Figma: "Not Published" state already drafted

Start with LLMProxyOverview.tsx, ServiceProviderOverview.tsx, LLMProxyDeploymentsCard.tsx, and ServiceProviderDeploymentsCard.tsx to trace the existing overview and deployment flows. Read the commented statusChipColor pattern in ProvidersList.tsx and the linked Figma designs before planning the shared UI. Done means the Deployments tab, deployment and publish badges, overflow actions, and publishing flow ship together without removing access to deployment or delete functionality.