Seems like if they opened up their API it would help. Wasn't that when everything started going downhill, when they started closing down third party apps?
Not following too many people is great, because the people you follow will curate the interesting stuff for you and re-tweet it. Stop following all media accounts for instance, maybe follow a couple reporters instead and you'll still catch all the interesting stories.
If un-following is socially awkward for whatever reason, mute people you follow or at least toggle off re-tweets for those that do too many.
I read my whole timeline most every day and it's great. It does take regular pruning though, like a plant in your garden.
Twitter should be making the experience valuable with as least work as possible from its users.