As far as I can tell, Buddhayana simply means you're a follower of the Buddha and aren't inclined to lock yourself into the doctrinal positions of the various schools... a non (or multi?) denominational Buddhist. The problem with Buddhayana tends to be the supersessionism inherent within Mahayana and Vajrayana, rendering the Theravada teachings as of lesser importance.
Buddhayana should not be confused with Buddhavacana (word of the Buddha), which is an attempt to use whatever means are at one's disposal, to find out what the Buddha actually taught and what he didn't... and follow only that which he taught! I consider this to be a legitimate approach to the Dhamma, but you need to be honest with yourself and have to agenda to push. Some people often use a Buddhayana style analysis to push their own pet theories.
As traditions/approaches/schools, both of the above are reasonably new and informal in their structure.
“I hope, Anuruddha, that you are all living in concord, with mutual appreciation, without disputing, blending like milk and water, viewing each other with kindly eyes.” (MN 31)
What is the final conviction that comes when radical attention is razor-edge sharp? That the object of the mind is mind-made (manomaya). (Ven. Ñāṇananda)
"If you have always believed that everyone should play by the same rules and be judged by the same standards, that would have gotten you labeled a radical 50 years ago, a liberal 25 years ago and a racist today." (Thomas Sowell)