That’s what retweets accomplish.

Youtube does this reasonably well; subscription page for channels you follow, and the home page for content discovery.

Make it explicit. A button that says "Show me something that I might like!" and supplies a list of tweets.

It could be a separate tab.

I normally use the realtwitter.com redirect to get just the tweets from people I follow. Occasionally I look at the regular feed and the other stuff can be interesting, but I’m also reminded of why I don’t use it by default.

You can still have search. And the people you follow can still post links (without those "cards") to content they find interesting - but in those cases you are in control. Nothing is forced into your feed.