So for none of them it was a fork. They all developed on the main branch, in the absence of centralized PR approvers everything gets merged.
In India this identity matters even more as the Indian constitution offers different rights to you based on your identity and you get to benefit from "unequal laws" hence everyone deeply gets sentimental about it.
One has to look how Islam treated the faiths around it when Islam obtained power. It was mostly genocide and iconoclasm at scale not seen before. Even acknowledgement of Jesus in the Koran is more to enable easier conversion into Islam. The concept of conversion for example existed in Islam.
Buddhism deals with far subtle and much more complex ideas like nature of reality. You can see a detailed discussion here: https://www.youtube.com/watch?v=vAZPWu084m4 (Note that the monk talking here is a Vedantist, talking with great respect and reverence to Buddha/Buddhism and on Buddha's birthday).
> Whatever the result, it's important to respect one's choice - if buddhists reject the place that modern Hinduism tries to fit them into, because they think the patch is not enough, that's their call and there is no point insisting otherwise.
I think that is a great analogy, but note that Hinduism has not tried to "fit buddhism" into anything, Hinduism only point out that Buddhism's fork is pretty much similar to many other forks of Hinduism which have not been identified as separate from Hinduism.
Neither Hindu nor Buddhist philosophy however cares about such distinctions or boxes as they are completely irrelevant to their objective, it is mostly for people to argue. (This is not true for Islam and Christianity, Islam denies you heaven if you havent accepted Islam before death, Christianity does the same, hence it matters deeply whether you are Christian or Muslim, mere believing in the God wont help.)
Overall "melting pot" mentality is a good thing and must be encouraged and a good thing. It helps people find common ground and love each other rather than needlessly quarrel and fight.
He (Swami Sarvapriyananda) has a lot of other interesting talks too, many of them about Advaita (Vedanta), which is the main focus of his work.
He is at the Vedanta Society of New York, which was started by Swami Vivekananda in 1894.
> This is how Hinduism evolves
It is indeed how religions evolve - literally by forking. Like Christianity forked Judaism (and was itself forked by Islam), Buddhism forked from Hinduism. Sometimes some patches are retrofitted on the original tree, sometimes trees diverge so much that it's hard to see the common root. Sometimes reformers go back to the original project, once their patches get somewhat accepted (a lot of anglicans after Vatican Council II...); sometimes they don't.
Whatever the result, it's important to respect one's choice - if buddhists reject the place that modern Hinduism tries to fit them into, because they think the patch is not enough, that's their call and there is no point insisting otherwise.