Age certainly has a negative effect on flexibility, but genetics has an even bigger effect. Although anyone may increase their flexibility, extremely flexible people are born that way.
Stretching is actually the wrong term to describe exercises used to increase flexibility. Muscles and ligaments don’t really stretch. What you are actually doing is teaching them to relax them while they are under tension, If they relax, they do not contract to protect themselves when placed under extreme tension, which allows them to lengthen (or stretch) more easily.
I’m over 60 and more flexible overall than my 30 something instructor. However, he is flexible where it needed for performing TKD techniques. Although he is not flexible overall, he is able to perform beautiful, powerful kicks with ease due to his performing the kicks daily.
Sometimes very flexible people are prone to injury since their joints are not as stable as less flexible people (you can’t have it all, with the good comes some bad). Therefore, the ideal is to have a flexible, powerful, and stable body.
Yoga specializes in total body flexibility; just pick yoga poses that stretch the muscles you need to perform the techniques you want to use. One of the best exercises for kicks is to hold to something stable and perform slow motion kicks, concentrating on using perfect form. As the body learns to perform the motions, it will relax and the kicks will naturally begin to get higher. When training, most students try to kick higher, harder, and faster, and they neglect form. A perfectly performed kick to the knee is much more effective than a sloppy kick to the head.