feat(gstack): share the Playwright bump, report the browser cache

Extract gstack_bump_playwright_if_unsupported from install-plugins.sh into
lib/gstack-playwright.sh and call it from update-all.sh too. A submodule
update no longer leaves the OS-support bump unapplied until the next
`make plugin` — that was BDR-029's open caveat.

The update helper never touches the submodule working tree: on failure it
prints git's own message and points at `make plugin`, and returns non-zero
so the existing `else warn` arm still handles it.

Add a read-only `Playwright browsers` section to doctor.sh: cache size,
which registered install requires each revision, and counts of unreferenced
directories and broken links. No pruning is written — Playwright's own
`install` already unions the required set across every registered install,
and all three installs here are live (rev 1228 for gstack + gsd-pi 1.61,
rev 1243 for gsd-pi 1.63).

Carries two latent-bug fixes from the moved code: the ostag capture exited
1 on every non-Ubuntu host and aborted the caller under inherited errexit,
and the bun calls had no timeout.

doctor.sh

@@ -18,8 +18,10 @@ REPO="$(cd "$(dirname "$0")" && pwd)"
 VERSION=$(cat "$REPO/version.txt" 2>/dev/null || echo "unknown")
 
 # Load shared detection library
-# shellcheck source=lib/detect-plugins.sh
+# shellcheck source=lib/detect-plugins.sh disable=SC1091
 source "$REPO/lib/detect-plugins.sh"
+# shellcheck source=lib/gstack-playwright.sh disable=SC1091
+source "$REPO/lib/gstack-playwright.sh"
 
 echo ""
 echo "═══ claude-config doctor (v${VERSION}) ═══"
@@ -115,6 +117,13 @@ fi
 
 echo ""
 
+# ── Playwright browsers (read-only report; NOT nested under gstack — 2 of
+# the 3 registered installs are gsd-pi, not gstack) ──
+echo "── Playwright browsers ──"
+gstack_browsers_report || true
+
+echo ""
+
 # ────────────────────────────────────────────────────────────
 # 3. Prerequisites
 # ────────────────────────────────────────────────────────────