But that's exactly how it is now.
I personally like the verified user option but I can understand how some wouldn't for reasons like the one you give. Frankly, I think Twitter's focus on growth is a big part of what got it into this problem in the first place. I'd rather see them focus on quality than growth and I think expanding the verified user program does that.
The measure-of-connectedness option is a little gentler and would probably fix a pretty big piece of the problem.
Quality of what? Discourse? Network connectedness? Ad-accuracy? (I'm nitpicking, but I'm curious what you mean.)
So the logical next question is how does Twitter shape how it is used? I think the answer to that is in part, let other people take care of that. In some ways, Twitter-the-platform is much more interesting that Twitter-the-product. My current favorite example is Nuzzle.
Unfortunately, Twitter is a $13 billion (and falling!) company. Investors think Twitter is Facebook Jr. and it really isn't.
One approach would be to expand their verified user program to people like me (ie the proletariat) and then have an option to switch the blacklist model of blocking people to a whitelist model of showing only tweets from verified users.
Or they could make a whitelist based on people I follow or that friends follow (or friends-of-friends etc...).
Or maybe just get serious with analytics and sentiment analysis and don't show me stuff that is likely a troll.