Fixes https://github.com/microsoft/playwright-mcp/issues/294
Moves the `schema.parse` call to the calling side of the handler, so we don't have to duplicate it everywhere.
Motivation: Keeping the readme up to date is a manual effort - this keeps it automatically up to date and prevents things like https://github.com/microsoft/playwright-mcp/pull/214 and other consistency errors in the future.
