Asana is indeed the most popular tool of Yoga in the world today. There are a couple of reasons, which I feel, I can attribute these to.
First of all, when Yoga spread outside India, it spread through certain great teachers like Sri. BKS Iyengar and Sri. Pattabhi Jois on the one hand, and certain disciples of Swami Shivananda on the other hand. Disciple of Swami Shivananda took more religious and a monk like approach. That is why you will see many of them engaged in Bhajans, Mantras and all these kind of practices. But students of Sri. Pattabhi Jois and Sri. BKS Iyengar took on a very physical approach because Sri. Pattabhi Jois and Sri. BKS Iyengar concentrated mostly on Asana practice. This is also because when Sri. Pattabhi Jois and Sri. BKS Iyengar studied Yoga with their Yoga Teacher Acarya T. Krishnamacharya when they were teenage boys in the 1930s and Arcarya T. Krishnamacharya taught them only Asana practices. He was of course teaching the king and the ministers in the Mysore palace the other techniques, but these students being young boys were part of the group that practiced mostly the Asanas practices.
Historically Sri BKS Iyengar moved away from Mysore when he was very young and Acarya T. Krishnamacharya also moved away from Mysore in the late 40s, so these students did not continue to have time with their masters to learn other practices of Yoga in detail. So, when their influence in the west increased, Yoga got established very much in the minds of people as Asana practices.
This is also because you could see that people like Sri BKS Iyengarji would give demonstrations to thousands of people showing amazingly and extraordinarily difficult Asanas which would “wowaha” the people and the audiences which got established very strongly in the minds of people as Yoga.
There is another reason I attribute to the popularity of Asanas. In modern society, the physical body is given intense focus. People want to develop the physical body, they want to look good, they want to look good in their profile pictures, Instagram pictures as well as in videos therefore the physical body gets a lot of attention. Asanas are very physical in nature in the eyes of many Yoga practitioners, eventhough classically Asana is not just a physical practice. This is another reason why the Asanas have become very popular in the minds of public as Yoga practices. That is also why you will see many people who retire from Gymnastics or dance like Bale and other things. they all get attracted to Yoga because it is very easy for them to do the physical postures.
These two are in my opinion the strongest reasons the why Asanas are the most popular tools of Yoga today. However, I sincerely urge every Yoga practitioner to look beyond Asana practice, which means necessarily to give up Asanas, but to transcend the Asanas and embrace the other wonderful, deep practices of Yoga. There are many wonderful benefits to be achieved. This will be a very good motivation for people to engage in the holistic practices of Yoga.
Dr.Kausthub Desikachar, Viniyoga Teacher