Run playground as part of CI

[wjhsf]

Recently, in #5187 and #5203, we inadvertently cause breakages that weren't caught by any unit or integration tests. However, they would have been caught if we had run the playground to check. So, let's do that!

We should somehow run the playground and check that "Hello World" and "Counter" are rendered on the page. Bonus points if we can do a click and check 0 -> 1. Also bonus points if we can run it with NODE_ENV=development and production. This implementation should explicitly avoid re-using the setup from the karma tests, because those don't catch what we're looking for (otherwise we wouldn't be here, in the first place).