Tag a subset of Playwright tests for running against live environments immediately after deployments to verify the deployment

[sarayourfriend]

Problem

We do not have any automated verification of live environment functionality after deploying them. We take for granted that things like configuration issues and such will cause problems sufficiently bad to surface in other ways (health checks, alarms, etc). That isn't necessarily the case though (#4703).

Description

Identify a subset of our Playwright tests that can be used against live environments to verify core workflows like, SSR search, CSR search, page metadata (head), and so forth.

These can be part of the regular suite, but focusable as a group (without needing to be in the same module) by a new tag on those tests, @deployment-smoketest.

We need to add a new dispatchable workflow to this repository to run these tests against an input that specifies staging or production. That new workflow should be dispatched at the end of the deployment workflows in the infrastructure repository, and the result should be reported to Slack.

Additional context

One idea coming out of the incident related to #4703 discussed by @zackkrida and me.

[sarayourfriend]

#5075 adds HMAC signing for requests in Playwright, and facilities for running the smoke tests after staging deployments. To fully close out this issue, more tests should get tagged with @deployment-smoketest to fully cover at least the critical paths of the application.