Re-enable the end-to-end job that verifies the web host

The end-to-end job that exercises the playground inside the web host is currently disabled
with a constant-false condition, noted as temporary while the smoke test was stabilised.

That is the check which would prove the web host works when it starts building against the
core at HEAD. Either it is re-enabled first, or the web host joins lockstep with weaker
verification than iOS and Android.

Blocks: #685.

Steps

• Establish whether the smoke test is now stable
• Re-enable the job, or record why it stays disabled and what covers it instead

Done when

The web host has an end-to-end check that runs, or a recorded decision that it does not
and what replaces it.