This is a stereotype of pagan religion that doesn't really hold true. Yes, the state was heavily involved in organising festivals, constructing temples and offering sacrifices - but the same holds true for the Abrahamic religions. And these state-sponsored events were not by any means the only expression of Akkadian polytheism. Worship was tied to the household and the individual as much as it was to temples and governments.
The idea that the pagan religions could only exist because they were upheld by the state is a bit of 19th century bigotry carried over into modern pop-science. It basically assumes that nobody actually believed in these religions and everyone was just in it because they had to.
I mean, that's precisely how the Esagila
was abandoned - when state support ran dry, the place gradually fell into obscurity.
It's not that the religions could not exist without state support, rather that they had surprisingly little staying power - like every other religion devoid of a larger organisation and in low-literacy periods - and of course, the discussion about whether 'people really believed' is equally shallow, a rebuke that is itself steeped in a superficially bad understanding of ancient religion.
In a polytheistic, pre-modern world, gods exist and only a few select people would dispute that fact. Nobody needs to 'believe' in them, not in a sense that our modern understanding, heavily shaped by Abrahamic religions, defines it. People by and large assumed gods existed and instead 'believed' in a lore of rituals and oaths, meant to bind and coerce the otherwise inscrutable and superior creatures into doing the invoker's bidding; hence you see people boasting about their chosen god's efficacy and belittling the others'. This dimension extends to mysteric cults, too - their rituals being so precious and special, only a few select ones are deemed worthy of partaking in.
So in this world, once the priesthood - which is by necessity tied to state power - is beaten, they also have been empyrically proven wrong, and once the residual sense of community fades, nothing is left to ensure the religion's survival.