Assyria is fine. Have you tried them? 2 siege towers backed up by composites destroy units and cities.
The Mongols are fine. Keshiks are amazing base units and upgrade from chariot archers. They get logistics stupidly fast. Then they get range. 5 Keshiks and a horseman can clear a deity Pangaea before they become obsolete, provided you prebuild chariots and tech to chivalry before education.
The Huns are not fine on deity, at least for me. Their UUs are amazing individually, but unless I get a free battering ram and a close enough neighbor to harass effectively, I cannot get enough horses and rams to attack before T50, and by then the AI can counter me effectively. Part of the problem in that rams cost 75 hammers, instead of 56. If I need 4 horses and 2 rams to make a go, which is not that many units, that's 350 hammers, or at least 20 turns that early in the game. I actually wish the battering ram and siege tower were switched, although this may hurt Assyria significantly. But classical era siege towers? I guess, but I don't associate Iron IIB with the classical era, although the game puts ironworking after that with currency. Nor do I think of the Huns as ancient era, seeing as they conquered in the 6th c. CE.