feat(code): add paginated github repo picker

[tatoalo]

Problem

The cloud gh repository picker only surfaced the first 50 repositories and depended on locally filtering a fully fetched
aggregate repository list. That made large integrations harder to navigate and behaved differently from the branch picker.

Changes

• add a paginated GitHub repository picker with Load more
• wire the task input and GitHub data source setup flows to use backend-backed repository search and pagination
• keep the existing full aggregate repository fetch for the other screens that still depend on it
• update the shared repository picker so it supports both local filtering mode and backend-backed search mode

[greptile-apps]

Prompt To Fix All With AI

This is a comment left during a code review.
Path: apps/code/src/renderer/hooks/useIntegrations.ts
Line: 99-101

Comment:
**`requestedLimit` never resets on search change**

The `useEffect` with an empty dependency array is a no-op — `requestedLimit` is already initialised to `REPOSITORIES_PAGE_SIZE` by `useState`. The intended reset (when the search query changes) is therefore missing. After a user clicks "Load more" (raising the limit to 100+) and then types a new search, the hook will request that elevated limit from the API instead of starting fresh at 50.

Change the dependency to `[deferredSearch]` to make the reset meaningful:

```suggestion
  useEffect(() => {
    setRequestedLimit(REPOSITORIES_PAGE_SIZE);
  }, [deferredSearch]);
```

How can I resolve this? If you propose a fix, please make it concise.

---

This is a comment left during a code review.
Path: apps/code/src/renderer/features/task-detail/components/TaskInput.tsx
Line: 553-556

Comment:
**Picker unreachable on first open**

When `isCloudRepoPickerOpen` flips to `true` and there is no cached data, `cloudRepositoriesLoading` (`isPending`) is `true` in the same render. This makes `isLoading = true`, causing `GitHubRepoPicker` to return the disabled "Loading repos…" button early — the `Combobox` is never mounted and the picker never visually opens. The issue exists identically in `DataSourceSetup.tsx`.

Because `useGithubRepositories` is only `enabled` when the picker is open, there is no pre-warmed cache, so the very first open (and any open after the 5-minute stale window) always triggers this path.

A straightforward fix is to only suppress the `isLoading` pass-through while the picker is closed and show loading state inside the already-open combobox (e.g. via a separate `isFetchingMore` / skeleton), or to enable the query slightly before the picker opens.

How can I resolve this? If you propose a fix, please make it concise.

_{Reviews (1): Last reviewed commit: "feat(code): add paginated github repo pi..." | Re-trigger Greptile}