Drop older asyncio workarounds for Python < 3.7

Since we've dropped 3.6, and 3.7 has A LOT of asyncio improvements ([here's a glance of some](https://docs.python.org/3/whatsnew/3.7.html#whatsnew37-asyncio) -- already a long list!), we might want to start replacing all 'workarounds' that catered to Python < 3.7 to things that are 'more elegant' (and definitely more up-to-date).

Not marking this for any particular version. But definitely making this a tech-debt.