Real browser test coverage (Playwright) #80

Note: The issue description may include details that are no longer accurate, as there have been many changes to the codebase since it was written. The underlying concept remains valid, though, which is to extend testing capabilities to include actual front end QA coverage.

Issue: The existing test suite (newtab.test.js, service-worker.test.js,
etc.) is solid jsdom-based unit coverage of logic, but nothing drives an actual
browser rendering the actual extension. The AGENTS.md manual QA checklist
("load unpacked, open a new tab, confirm cards render...") is currently a
human/agent-eyeball process regardless of how it's documented — that's the real
gap, independent of the AGENTS.md wording itself.

Ask: Set up Playwright test scaffolding capable of loading the unpacked
extension in a real Chromium instance and driving the core flows end-to-end:
new-tab renders a card, Settings open/close (including the Task 1 fix above),
ZIP/location submission, and the explore-mode flow (explore → show another cat
→ back to my area). Generate whatever config/fixtures/tooling is needed so this
can be run headlessly by an AI agent (yourself or another Claude Code session),
not just by a human locally. Once in place, update the AGENTS.md manual QA
section to reflect the new split — which parts are now covered by Playwright
vs. what (if anything) still needs a human/agent eyeball.

Testing required: N/A (this task is the test infrastructure) — but do
run the new Playwright suite yourself before considering the task complete.