Perhaps a separate plugin?
Daily and weekly digests.
That's an interesting one. The Daily Digest was the main feature I once installed the TT for :-)
I think there are two possible routes:
To me when swapping to the new version (3.x) of TT it wasn't clear that TT will populate the wp_post table with tweets (using the aktt_tweet post-type), and that enabling "make posts from tweets" would just make a "real" post. So initially I had the misunderstanding that only "posted" tweets would appear in the wp_post table.
Since I use(d) TT to keep an archive of may tweets, there's no need to publish additional posts when a new tweet has been fetched. It took me a while to get that.
This goes slightly off-topic now, but why did you decide to insert the fetched tweets directly into the wp_posts table? Naivly looking from the outside, an aktt_tweets-table seems cleaner?
There are some notes on the upgrade in my blog: http://alexking.org/blog/2012/10/17/twitter-tools-data-upgrade-process