I'd say - and I should point out I only dare speak from the Chinese side of this, but I'll be thoroughly unsurprised if it markedly differed among the other ancient empires - that there is one major point that might help explain your question, and that is: there is/was a fundamentally different outlook between a state religion of a regional superpower, and that of an "underdog" faith.

So in the case of China (and Rome, etc, etc), the core sociocultural outlook was that "We Are Civilization." And to that effect, the way they dressed, groomed, thought, wrote, spoke, prayed, and believed were necessarily also the markers of civilized thought. The expectation was largely that peripheral states would - by sheer dint of the empire's own innate awesomeness - naturally seek to Sinicize/Romanize, etc, to become more civilized and closer to "good human-ness. When your worldview stems from such a place, there really isn't much of a drive to go "spreading the Good Word of the Dao" or the like... people come to you and seek to emulate your thought, you're not having to go door-to-door about it. There just isn't the motiviation... anyone close by too stupid or animalistic to realize that your way is obviously the better way... well, they will either come around, pay a tax, or be exterminated; bicycles and suit not required.

Contrast that, then to the said "underdog" faiths that you hint - the one in the West is obvious enough: that Judean Cult of the Carpenter. But China, too, had its own little strange, foreign upstart - and one that ultimately came to, well, evangelize itself into predominance not only in China, but also almost the entirety of East and Southeast Asia: Buddhism.

Buddhism never really had "legs" in its own homeland - oh sure, there was Ashoka's Empire and his stone pylons, but that died with him... rather, on the whole, it had to travels to very long and difficult journeys to find its "promised land": one over the vast, empty stretches of the Silk Road, and the other via the tides and currents of the Indian Ocean trade routes. Since I'm not in any way shape or form qualified to talk about the latter, I'll focus on the former...

Buddhism came to China in the early stages of the Han Dynasty [ca. 206 BCE - 220 CE], but it wasn't some great bum-rush. It came in fits and starts, and didn't really find a longterm toehold in the empire until it was broken into pieces - amid the ~400 year civil war that was the Period of Disunion [220-581 CE]. Even then, there was quite the intellectual furor over whether or not this popular religion was compatible with traditional Chinese values. There had been ongoing (and occasionally quite violent) dialogue between the camps. From "The Disposition of Error", ca. 500 CE:

If Buddhism is the greatest and most venerable of ways, why did the great sages of the past and Confucius not practice it? In the Confucian Classics no one mentions it. Why, then, do you love the Way of the Buddha and rejoice in outlandish arts’? Can the writings of the Buddha exceed the Classics and commentaries and beautify the accomplishments of the sages?

to which the buddhist replied:

All written works need not necessarily be the words of Confucius. To compare the sages to the Buddha would be like comparing a white deer to a unicorn, or a swallow to a phoenix. The records and teachings of the Confucian classics do not contain everything. Even if the Buddha is not mentioned in them, what occasion is there for suspicion?

That certainly did not end the debate within China. And though the subsequent Tang dynastic rulers did largely embrace Buddhism, that it likely owed in large part to their non-Chinese background (half-Turkic) which had accepted the faith much earlier and more wholesale than their Yellow River Valley brethren. Though the Tang ruling class were often quite devout, that in no way stopped their ministers and officials from speaking out passionately against what they saw as a creeping, foreign cancer. So states the Chancellor Han Yu in 819 CE:

Your servant begs leave to say that Buddhism is no more than a cult of the barbarian peoples spread to China. It did not exist here in ancient times. Now I hear that Your Majesty has ordered the community of monks to go to greet the finger bone of the Buddha, and that Your Majesty will ascend a tower to watch the procession as this relic is brought into the palace. If these practices are not stopped, and this relic of the Buddha is allowed to be carried from one temple to another, there will be those in the crowd who will cut off their arms and mutilate their flesh in offering, to the Buddha. Now the Buddha was a man of the barbarians who did not speak Chinese and who wore clothes of a different fashion. The Buddha’s sayings contain nothing about our ancient kings and the Buddha’s manner of dress did not conform to our laws; he understood neither the duties that bind sovereign and subject, nor the affections of father and son. If the Buddha were still alive today and came to our court, Your Majesty might condescend to receive him, but he would then be escorted to the borders of the nation, dismissed, and not allowed to delude the masses. How then, when he has long been dead, could the Buddha’s rotten bones, the foul and unlucky remains of his body, be rightly admitted to the palace? Confucius said: “Respect ghosts and spirits, but keep them at a distance!” Your servant is deeply ashamed and begs that this bone from the Buddha be given to the proper authorities to be cast into fire and water, that this evil be rooted out, and later generations spared this delusion.

Nevertheless - and in spite of a largescale resurgence during the Song of a revitlaized - one might even say "Neo" - Confucianism... Buddhism ultimately won out in the hearts and minds of the people, both within China and along its periphery. Those doctrinally accepted imperially-sanctioned religions are all well and good when an empire is on the rise and can actually expect its neighbors to come a flocking to walk, flap, and quack like itself... but that self-assuredness and pointed disdain or marketing itself too openly has historically run the risk of it losing ground amid a startup, grassroots faith that preaches and fundamentally more relatable and egalitarian message.