Split out get_auth_request(), so that we can re-use it for other
scenarios.

Remember the last exception and print it in the timeout failure message, for
better debugging.

If they crash, removal makes the logs disappear, which makes debugging
hard.

`make clean` removes them anyway, so there is no risk of permanently
leaking them.

This is conceptually broken -- the browser should decide when it wants
to upgrade a connection to websocket, and we should just pass that
through (like we do with `Upgrade:` already). This is also what the real
3scale does, so more faithfully reproduces its behaviour.

This also helps to avoid aio-libs/aiohttp#4581