Hinduism did centralize in a way during the Muslim invasions. Prior to them, Hinduism was rather more diverse, and afterwards, the main Hindu school became Advaita Vedanta, which preaches that the soul and God are one. This is because Advaita Vedanta is the most flexible of the Hindu schools and therefore followers of other schools could follow Advaita without having to drastically alter their beliefs. This isn’t to say there aren’t divergent beliefs, like those of the Chaitanya Vaishnavites or the Lingayati, but most Hindus today are followers of Advaita Vedanta
This is certainly not what you mean, of course. What you’re asking for is impossible. Certainly, you could get some Hindus centralizing their faith, but this would be nothing more than a splinter group, a special form of Hinduism. It’s impossible for these centralizations to be accepted by all, because they would, by their very nature, deemphasize some elements of Hinduism another group may want emphasized. No “central” holy text could ever exist, because the Vedas already serve that purpose and attempting to replace it would be heretical - by definition, Hinduism is a religion that has the Vedas as the central holy text. Attempting to strip down Hinduism to the Vedas would also fail because they can be interpreted in many ways. Arya Samaj tried that, but it turned out most Hindus disagreed with their interpretation of them as preaching a monotheistic anti-idolatry faith. Any interpretation of the Vedas would come off the same. Also, Hinduism is by its very nature vague and somewhat contradictory. These attributes have Hinduism an ability to adapt that helped it to survive, with Muslim and even Christian elements being incorporated. By centralizing Hinduism, you are removing its dynamism, which is likely a detriment to its success.
I would argue there were many attempts to create a centralized faith based on Hinduism. All of them failed to become the religion of a majority of Hindus for the reasons above. The only thing I can think of that would change this is some centralized empire attempting to stress its own form of Hinduism against others, but even this would fail, judging by South Indian empires attempting to stress alternatively Vaishnavism and Shaivism, even going as far as destroying temples of the “other”, but nevertheless failing all the same.
The Hindus never called the religion by that name till the Muslims came. The common term used to denote it was 'Sanatan Dharma' or 'Eternal Faith'.
Sanatan Dharma is a neologism, largely promoted by nationalists who think “Hindu” is too foreign. No, the pre-Muslim Hindus referred to their religion as
Āstika, in opposition to
Nāstika schools like Jainism and Buddhism.