Closes https://github.com/microsoft/playwright/issues/32159. I
originally set out to enable Strict Mode for our React UI, but found a
way better thing: Enabling the lint rules we had already installed!
`eslint-plugin-react` is already in of our `package.json`, and this PR
enables it and fixes some of the reported issues. Most of them are
around the `key` prop which is mostly about performance, but there's
also fixes for misspelled `data-testid` props.
