I think people confuse two different kinds of stretching. There's stretching the muscles, which used to be thought useful in enhancing performance and reducing injury (which turns out to be false), and there's stretching the connective tissue for increased flexibility and range of motion, which is useful to varying degrees depending on the activity. You're going to see dancers and gymnasts doing a lot of stretching, but it may not be very useful for a football player.

Stretching connective tissue does cause micro-tears, and it does loosen it up the tissue. That's not necessarily a bad thing. We want connective tissue to be strong when it serves our purposes, and pliant when that serves our purposes. When you injure a joint, you will build up connective tissue. It's a natural bracing to allow healing and reduce further injury. But that new connective tissue (aka scaring) then becomes an impairment, and you have to stretch and break it down to get a return of necessary range of motion. Likewise as you age, and sit on your butt most of the day, you loose beneficial flexibility in your hips, your back, your shoulders, and so on. Having flexibility in the lower back and in the neck are areas that I think are very beneficial for bicyclists, and probably the hips too. It allows you to hold a good riding position more comfortably, have a more effective pedal stroke, and look around as necessary without having to turn your entire body.

It's also apparent that some people feel much better after stretching, and some people find it a waste of time, or too painful to reach any potential benefit.

BTW, I know you can't selectively stretch either muscles or connect tissue. My point is that the focus of useful stretching should be to increase flexibility with minimal trauma to the muscles. It's probably impossible to do really deep stretching without some muscle breakdown. That's why I wouldn't jump on my bike after a good long yoga session, but I know that yoga session is going to make me feel better and, in the long run, perform better.