Avatar is a very Vaishnava idea. It solves too many problems in one go. There are three aspects that are essential to be considered an Avatar.
-As the word itself connotes, it is a descent of higher consciousness amidst the world play, avatarana. In this one line, an important aspect of the world would be rendered as an illusion conjured by Mind as per Buddha. Far from Play it is considered as unreal. So is all experience that springs forth from the world experience is unreal. Vaishnava view on the contrary never disposes of the world. There might be varying views of worlds oneness with the Ultimate Truth. But the world is never relegated to being Unreal. So it is unwise to force-fit Buddha into this scheme of things.
-Avatar descends not just for the sake of Lila, but there is purpose that of upholding Dharma. Vaishnava view again emerges from the fact that Vishnu is the force that sustains the manifestation. The unseen yet abiding principle force that truly sustains this manifest order is Dharma. This Dharma is very different from Buddha’s dhamma. Buddha exhorts his followers to recognize the unreality of the world. It is in this recognition that they can alleviate themselves of all suffering. So there is a principle contradiction.
-Another nuanced point is to understand the notion of the Avatar. It is in Srimad Bhagavad Gita that we come across this proclamation by Bhagavan himself. This vision in a way emerges amidst churning discussion about the unbridgeable gulf between Akshara and Kashara purusha. Latter being the Purusha which is involved in the mutable world bubbling with Pravritti and former being the unmanifest indifferent and unaffected witness. Realizing the Akshara state in its essence naturally pushes for the abandonment of the Kshara. This is called Nirvana. Closer to Buddhas dhamma. This paradox is reconciled by Bhagawan by pointing to the transcending truth into which mutable and immutable can be ushered in without conflict, such ultimate all-embracing power is called as Purushottama. This is great, but still doesn’t serve the complete purpose. As it would still leave the world of lower prakriti of Kshara full of ignorance and no scope of any enlightened action. You see Srimad Bhagavd Gita is a text for enlightened action. Bhagawan by introducing the principle of Avatara establishes an ideal that there is a possibility to remain in the world order and yet rooted in param sathyam and act from this pedestal. If Arjuna were to ask for example, Bhagavan could simply say, look at me I am the example. Abandonment of the world is no choice Oh Arjuna, enlightened action is the only way to uphold Dharma. Buddha would never ever say that.
So its best we abandon this idea of Buddha as part of the Avtarhood scheme and learn from him on his own terms. He is a stunning example for persisting to figure out the cause for Dukham. That he found out that idea of Self is the source for this cause might not be acceptable to us. But as a civilization we allowed so many contrarian ideas to inspire us. In that spirit we can celebrate Buddha and refrain from force-fitting him into the scheme of things where he doesn’t belong.