Wherever Buddhism spread, it generally adapted itself to the dominant social order already established in the region, whether that meant the centralized imperial bureaucracy of China, the militarized feudal structures of Japan (not much different of a caste system, even having their own untouchability), or the caste-based organization of Hindu society in India.
It also tended to accommodate the existing religious and cultural customs of local populations. One notable exception often mentioned is the suppression of the practice of burying widows alongside their deceased husbands in Mongol territories under the influence of the third Dalai Lama. In most cases, however, Buddhism primarily incorporated members of those societies into the monastic community and directed them toward Buddhist discipline and practice. Even in what is often described as “caste-bound India,” the Buddhist emperor Ashoka, despite taking the significant step of restricting animal slaughter on certain days, did not attempt to dismantle the caste system itself.
That's why more than 80% of the hundreds of men Siddartha recruited, were from the upper castes, more than 40% were Brahmins and that the Buddha himself was a proud Ksatriya (and the Sangha even proudly flaunted that status, as seen in the controversy over who would keep his ashes) and prophetized that his sucessor would also be from a upper caste.
There is nothing that justifies the Ambedkarite narrative that, on a social level, Buddhism was a reformist movement aimed at abolishing caste. On the contrary, in many cases it was tolerant of it and greatly benefited from it.
“What the noble ones say is the truth, what the others say is not true. And why is this? The noble ones understand things as they are, the common folk does not understand. Furthermore, they are called noble truths because they are possessed by those who own the wealth and assets of the noble ones. Furthermore, they are called noble truths because they are possessed by those who are conceived in the womb of a noble person.” - Abhidharma Mahāvibhāṣa Śāstra
"And why, monks, did the Bodhisattva behold the country of his birth? Because a bodhisattva is not born in outlying lands where people are as stupid as sheep, with dull faculties, ignorant, and incapable of distinguishing right from wrong. Rather a bodhisattva is born in a central land. And why, monks, did the Bodhisattva behold the family of his birth? Because a bodhisattva is not born into an inferior family, like a family of outcastes, flute makers, cartwrights, or servants. A bodhisattva is only born into one of two families—a priestly family or a family of the ruling class. When the priestly families are dominant in the world, the bodhisattva is born into a priestly family. When the rulingclass families are dominant in the world, the bodhisattva is born into a ruling-class family. Thus, monks, at this time the ruling-class families were dominant in the world, so bodhisattvas were born into such families. " (Lalitavistara Sutra, c. 3)
Since you are a foreigner and new to this topic:
1. There is no Caturvarna in Buddhism and wherever you find these terms mentioned, they only serve a rhetorical function that's not central to Buddhism. If it was, all the Buddhists across the Buddhist countries would be practising it.
2. Shantideva, who is the author of the quintessential handbook for Boddhisattvas, not once mentions anything about caste.
3. Buddhists are not obliged to answer or provide explanations to an inferior religion. You take our answers or leave it.
4. You will not understand the sublime wisdom of the Buddha. This is made clear in multiple texts including Parinirvana Sutra.
Buddhadharma is SUPREME and those who drink the sweet nectar to take refuge in the Buddha are immune to the howls of the jackals or the whispers of the demons.