ci: reliably install WebKit for mobile Playwright e2e

[PAMulligan]

Problem

The pnpm e2e workflow fails on every run with:

browserType.launch: Executable doesn't exist at /home/runner/.cache/ms-playwright/webkit-2272/pw_run.sh

The 3 mobile tests fail while the 7 desktop (chromium) tests pass. The mobile project emulates iPhone 13, which Playwright runs on WebKit — not Chromium, despite the project being named chromium-mobile.

Root cause

A cache-poisoning foot-gun in e2e.yml. The browser-binary install was gated behind if: cache-hit != 'true', while the cache-hit branch ran only playwright install-deps (system libraries, no binaries). Git history shows the project started chromium-only (33a6fc0) and later added webkit (93b28bf). Once the cached ~/.cache/ms-playwright was missing the current WebKit build, the cache-hit path could never repair it — the binary install was permanently skipped — so the failure persisted on every run.

Fix

• Install browsers unconditionally with playwright install --with-deps chromium webkit. The command is idempotent and only downloads what's missing from the (cached) dir, so a partial/stale cache self-heals instead of being skipped forever.
• Key the cache on the resolved Playwright version, which pins the exact browser builds — a Playwright upgrade now auto-invalidates the cache.
• Rename chromium-mobile → mobile-safari. It has always run on WebKit; the misleading name is what led CI to install chromium only to begin with.
• Add webkit to the local e2e:install script so local devs running the mobile spec don't hit the same wall.

Verification

Full e2e suite run locally — 10 passed across chromium-desktop and mobile-safari:

Running 10 tests using 8 workers
  ✓ [mobile-safari] › opens as a bottom sheet with a scrim and a drag handle
  ✓ [mobile-safari] › tapping the scrim closes the bottom sheet
  ✓ [mobile-safari] › send a message and receive a reply on mobile
  ✓ [chromium-desktop] › ... (7 tests)
  10 passed (11.8s)

No widget source code changed — only CI config, the e2e project name, and the install script.

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