Central Asia, I will note, had some Hindu influences such as that Shiva was very popular there, with a form of him known as "Oesho" apparently being a major god in the pantheon of the Central Asia-India straddling Kushan Empire. I guess the fact that Shiva is a mountain god was very appealing to them. Assuming that an empire straddling the Central Asia-India frontier exists for centuries, you could easily get Shiva and other Hindu gods as major Central Asian gods.
Some forms of Hinduism did indeed proselytize, though most of those forms of Hinduism emerged later. For instance, Chaitanya Vaishnavism converted people through wandering ascetics (vairagis), as
I've noted before. And even later than that, in the late nineteenth century, the Vedic revivalist sect of Arya Samaj was also big on conversion. What these movements have in common is that they were organized sects, unlike traditionalist Hinduism which is hardly organized.
So, what you really need is a rather organized form of Hinduism to emerge (maybe out of Bhakti Hinduism, as many such sects did emerge) and then spread its new "fundamental truth" across the world.
I think what differentiates (traditional) Hinduism from Abrahamic religions is that it isn't a very organized religion. It has many, many, holy texts and hymns, some of which contradict one another, whereas the Abrahamic religions have a single holy text. Hinduism itself is the syncretism of the Aryan religion brought after the migration of Aryan peoples with the native religion of the Indians, and then the further syncretism of these syncretisms, which is why (for instance) Shiva's wife has the extremely different forms of Parvati, Kali, and Durga or that Shiva is simultaneously the lord of animals, a mountain god, and a dancing god. This has resulted in Hinduism being extremely flexible to an extent that other religions are not.