feat(sessions): remove stale-conversation cost notice

[charlesvien]

Problem

When PostHog staff returned to a large, idle conversation, a cost notice appeared warning that the prompt cache had likely expired. It started as a blocking prompt over the composer; this PR first made it a small non-blocking banner, but the conclusion from discussion is that the notice shouldn't exist at all.

As a non-blocking banner it's just visual noise — it doesn't change behaviour, and it's only shown to staff. And compacting isn't the right recommendation: you pay the same uncached full-price input cost on the next message whether or not you compact, and a normal message re-warms the cache for subsequent turns regardless. So the "compact" call-to-action was misleading.

Changes

Remove the stale-conversation notice/gate entirely:

• Deleted the banner component (+ stories), the gate hook, and the gate store (+ test).
• Removed the shouldWarnStaleCostlyConversation / extractLastActivityAt helpers and their thresholds/tests from core.
• Removed the STALE_CONVERSATION_GATE_CHOICE analytics event and its properties.
• Removed all wiring from SessionView.

Net ~630 lines deleted; no behavioural change for users (the notice was staff-only).

How did you test this?

• pnpm --filter @posthog/core --filter @posthog/ui --filter @posthog/shared typecheck — clean
• pnpm --filter @posthog/core test contextUsage — passing
• biome ci . — clean (exit 0)

Automatic notifications

• Publish to changelog?
• Alert Sales and Marketing teams?

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Created with PostHog from a Slack thread

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[github-actions]

React Doctor found no issues in the changed files. 🎉

_{Reviewed by React Doctor for commit f9325f5.}

[greptile-apps]

Comments Outside Diff (1)

1. packages/ui/src/features/sessions/components/SessionView.tsx, line 356-359 (link)

   The analytics event still records "continue" for what is now a dismiss action in the UI. If these events are queried to understand user intent (compact vs. continue vs. new-session), the label will not match the new banner vocabulary — future readers of dashboards will see "continue" where the user actually clicked "✕ dismiss". Renaming to "dismiss" keeps the tracking consistent with the redesigned UX.

_{Reviews (1): Last reviewed commit: "fix(sessions): make stale-conversation c..." | Re-trigger Greptile}

[greptile-apps]

_{Reviews (2): Last reviewed commit: "feat(sessions): remove stale-conversatio..." | Re-trigger Greptile}

[greptile-apps]

_{Reviews (3): Last reviewed commit: "feat(sessions): remove stale-conversatio..." | Re-trigger Greptile}

[stamphog]

Gates denied this PR because it exceeds the automated review size ceiling (673 lines, 9 files). While the change is mostly deletions (removing the stale-conversation cost notice feature), the scale requires a human reviewer to confirm no dependent code paths were missed. Request a review from a teammate before re-requesting auto-approval.