I'm bumping this thread.
Like others said it's not convenient. Imagine storing hundred or thousand of tweet,
we can not do an API query for each one, just to be sure the avatar still works.
Facebook does actually provide the same feature
You just have to have the user_id
Even for twitter itself it's not good, if we display broken avatar, your product feels broken too.
This 3years issue just tell developpers/salesman that twitter is a not a good things to use,
because we can not unsure a reliable way to display tweets (not counting the js sdk that works great)
Which is a shame, because twitter is a wonderfull plateform to discuss and exchange.