Twitter is not email. You don't have to read all the tweets. The perfect Twitter client will show me which feeds have new tweets and then when I get to the top, assume I've read them all.
Hey, cool, that's exactly what Tweetie does.
A Twitter client is for client side Twitter use. You can't get push notifications from a Twitter client. In order to get push notifications, you would need a twitter service. Tweetie is not a service, it is a client. Maybe Twitter will come out with its own push service. That way all these developers won't have to run a server that checks all of the accounts of all people using their client and sends push notifications.
Until then, Boxcar works fine.
My tweets are garbage? Probably shouldn't follow me, then. Don't you love how Twitter works that way? If you don't like someone's tweets, you don't have to see them? It's great.