chore(ci): add dedicated TypeScript typecheck workflow and root typecheck script

[Harsh23Kashyap]

Problem

The PR-time Lint and Test workflows in .github/workflows/ run yarn lint and yarn test respectively, but neither invokes tsc on the web package. The only place tsc runs for the web package is as a side-effect of next build inside the pr-gate.yml Docker-image build job, which is slow, shared with image building, and not the focused signal that developers watch during iterative review.

The other workspaces (backend, db, schemas, shared) all have tsc in their build scripts, so they are type-checked when those packages are built. web is the only workspace without a dedicated typecheck path.

Impact

• A PR that introduces a TypeScript compile error in packages/web is only flagged once the slow Docker build runs in pr-gate.yml. A dedicated tsc step on a focused workflow would surface the same error in seconds.
• Local developers have no yarn typecheck shortcut to type-check all workspaces; they must either run tsc per package or trigger a full next build.

Proposed solution

This PR is the first step of a two-step plan:

1. This PR (scope-cut): Add a typecheck script in the root package.json that runs tsc --noEmit in each workspace that ships TypeScript (web, backend, db, schemas, shared), using the same topological ordering as the existing test/lint scripts. Plus a per-package typecheck script (mirroring the existing test/lint patterns). Local developers can now run yarn typecheck from the repository root.
2. Follow-up PR (separate scope): Add a new CI workflow (.github/workflows/typecheck.yml) that runs yarn typecheck on PRs. This depends on first fixing latent type errors that yarn typecheck surfaces on current main (see Future work below).

Scope of this PR

• 1 root package.json change (additive).
• 5 per-package package.json changes (additive, one typecheck script each).
• No source code changes. No new dependencies. No new CI workflow.

Acceptance criteria

• yarn workspace @sourcebot/web typecheck exits 0 (with the caveats below).
• yarn workspace @sourcebot/backend typecheck exits 0.
• yarn workspace @sourcebot/db typecheck exits 0.
• yarn workspace @sourcebot/schemas typecheck exits 0.
• yarn workspace @sourcebot/shared typecheck exits 0.
• yarn typecheck from the repository root invokes each workspace's typecheck in topological order.

Future work (out of scope here)

The current main has latent type errors that surface when tsc --noEmit runs against each workspace in isolation:

• packages/web test fixtures reference fields that exist in DB migrations but were not added to the schema.prisma model (e.g., UserToOrg.lastActiveAt from migration 20260702000001, UserToOrg.suspendedAt/scimExternalId from 20260702000000). Fixing this requires either (a) updating schema.prisma to declare those columns, or (b) re-running prisma generate, depending on intent. Either way, that is its own scope.
• packages/db had a stale Prisma client before this work; running yarn prisma:generate locally resolves the immediate AgentSkill* errors. This is documented as a prerequisite for the follow-up CI workflow PR.

A follow-up PR will: (1) fix the schema-drift in schema.prisma, (2) update test fixtures to match, (3) add the .github/workflows/typecheck.yml CI workflow introduced in this issue's proposed solution. Tracking in this issue so the work isn't lost.

Backward compatibility

None. Pure additive tooling. Existing build/test/lint scripts unchanged.

[Harsh23Kashyap]

Picking this up. The local tooling half — root yarn typecheck plus per-package typecheck scripts — is on branch Harsh23Kashyap:chore-ci-typecheck-workflow. Cross-fork PR-create API is rejecting my head today (same GitHub-side quirk we hit before on this fork), so PR opening falls back to the web URL: https://github.com/sourcebot-dev/sourcebot/compare/main...Harsh23Kashyap:chore-ci-typecheck-workflow?expand=1

The PR scopes the tooling half of the issue and explicitly defers the CI workflow half — adding .github/workflows/typecheck.yml requires first fixing the schema-drift in packages/db/prisma/schema.prisma (UserToOrg.lastActiveAt / suspendedAt / scimExternalId added in migrations 20260702000000 and 20260702000001 but not declared) and the ~30 test-fixture types in packages/web that reference those columns. That is its own scope.

Will not post a daily nudge; will leave a single follow-up comment on this issue if/when maintainers want to sequence the schema-drift fix in a separate PR.

[Harsh23Kashyap]

PR opened: #1438

Opened on branch Harsh23Kashyap:typecheck-workflow (had to switch from Harsh23Kashyap:chore-ci-typecheck-workflow because the cross-fork PR-create API on this fork was rejecting the Harsh23Kashyap: ref shape for cross-fork heads; simpler bare ref worked).

This is the local-tooling half of the issue. The CI workflow half (the .github/workflows/typecheck.yml file) is intentionally deferred to a follow-up PR — see the PR body for why. No daily nudge will be posted on the PR.