Let's distinguish between two things. It is clearly not true that the word Vedanta is more popular than the word Yoga; in modern times even white people in America know the word Yoga, whereas almost no one know what Vedanta is. What is true, however, if you look at any given major sect of Hinduism, almost all of them subscribe to the Vedanta school, regardless of the ignorance of individual members. Concerning Patanjali's Yoga school, in contrast, while it is the case that the Ashtanga Yoga procedure given in the Yoga Sutras is quite popular especially in recent times, that does not indicate the popularity of the actual philosophical commitments of the Yoga school; for instance, the Yoga school believed that the supreme being is the efficient cause but not the material cause of the Universe, a belief that most major sects of Hinduism disagree with. (There are a few sects that do believe that the supreme being is the efficient cause but not the material cause, like the Shaiva Siddhanta sect, but they disagree with the Yoga school for other reasons.)
Now without further ado, let me illustrate the wide acceptance of the Vedanta school, by enumerating major sects of Hinduism and how they view the Brahma Sutras, the defining text of the Vedanta school:
Accepts Vedanta:
- Adi Shankaracharya's Smartha sect subscribes to Adi Shankaracharya's Brahma Sutra Bhashya, which you can read here and here, and which preaches a Vedantic philosophy called Advaita.
- The Sri Vaishnava sect subscribes to Ramanujacharya's Sri Bhashya, which you can read here, and which preaches a Vedantic philosophy called Visistadvaita.
- The Madhwa sect subscribes to Madhvacharya's Brahma Sutra Bhashya, which you can read here, and which preaches a Vedantic philosophy called Dvaita.
- The Gaudiya Vaishnava sect subscribes to Baladeva Vidyabhushana's Govinda Bhashya, which you can read here, and which preaches a Vedantic philosophy called Achintya Bhedabheda.
- Nimbarkacharya's sect subscribes to Nimbasrkacharya's Vedanta Parijata Saurabha, which you can read here, and which preaches a Vedantic philosophy called Dvaitadvaita.
- The Shrauta Shaiva Siddhanta sect, to which Srikantha Sivacharya belonged, subscribes to Srikantha Sivacharya's Srikantha Bhashya, which you can read here, and which preaches a Vedantic philosophy called Shivadvaita or Shiva Visistadvaita.
- The Pushtimarga sect subscribes to Vallbhacharya's Anubhashya, which you can read in Sanskrit and Hindi here (I haven't found an English translation), and which preaches a Vedantic philosophy called Shuddhadvaita.
- The Ramanandi sect subscribes to at least five commentaries on the Brahma Sutra, which I discuss in my question here, which don't seem to have been translated into English, and which preach a Vedantic philosophy called Visistadvaita which is likely quite similar to Ramanujacharya's philosophy of Visistadvaita.
- The Swaminarayan sect subscribes to Muktanand Swami's Brahma Sutra Bhashya Ratnam, which you can read in Sanskrit and Gujarati here (I haven't found an English translation), and which preaches a Vedantic philosophy called Panchatattva Visistadvaita.
- The Veershaiva sect subscribes to Sripati Pandita's Shrikara Bhashya, which you can read in Sanskrit here (I haven't found an English translation), and which preaches a Vedantic philosophy called Shakti Visistadvaita, which I discuss here.
- The Vaikhanasa sect subscribes to Srinivasan Dikshitar's Lakshmi-Visistadvaita Bhashya, which I haven't found in either Sanskrit or English as I discuss here, and which as the name suggests preaches a Vedantic philosophy called Visistadvaita.
Rejects Vedanta:
- The mainstream Shaiva Siddhanta sect rejects the Vedanta school, as can be gleamed from the quote in my question here, and also from Sadyojyoti's Paramoksha Nirasa Karika, which you can order here. (It's interesting to note, by the way, that Sadyojyoti lived a century before Adi Shankaracharya, so he was writing at a time when Advaita was not yet popular in the Vedanta school, as is clear from his characterization of the Vedanta school.)
- The Saiva Siddhanta Church rejects the Vedanta school, as you can see from the personal correspondence quoted in my answer here.
- Abhinavagupta's Kashmiri Shaivite sect rejects the Vedanta school.
- Bhaskararaya's Sri Vidya sect rejects the Vedanta school.
- The mainstream Lingayata sect rejects the Vrdanta school.
