To enable Family Library Sharing, first be sure you have Steam Guard security enabled via Steam > Settings > Account in the Steam Client. Then enable the sharing feature via Settings > Family, (or in Big Picture mode, Settings > Family Library Sharing,) where you'll also authorize specific computers and users to share.
.... you've to do this from account that owns the game.
They have this popup to request sharing - which does not work (i think they're also aware of bug)
You actually can use ctrl+f5 to reload images/scripts and stylesheets. Works 99% of time.
This tho (as well as clearing your cache from settings) will reload the image for you only. All other site users will probably see the old image as it is cached on each of their browsers (until it decides that it should reload image(s)).
Since custom avatar images name is your ID - only thing to do here is save somewhere value which is included when rendering image (as: 12345.jpg?=some_value) when some_value could be the timestamp for example and update it when new avatar gets uploaded. (or find another way to name avatar files)
Another way would be to inject there some value whitch (same way) - for example date formatted (ddMMyyyy) - which would make u free of need to create another column for users. It would then reload avatars for everyone once a day.