There are several threads to this.
1) We don't actually have many surviving writings from thousands of years ago (surprisingly), and also there weren't necessarily a huge amount of people writing anyway (not many people + low literacy rate). So there would be a limited quantity of material doing what you describe, and a limited amount of it would have survived, meaning that just because we don't have much now doesn't mean it didn't exist.
2) Every other god was the spaghetti monster. But you have to consider that the way religion was perceived back then (outside of Christianity/Judaism) was very different to our view which has been shaped by Christianity for 1000~1500 years.
3) People did write about how some elements of Christianity were bunk, especially classical authors, who describe stories of men using various tricks which were claimed to be magical or similar to miracles in the bible. Now these aren't all specifically targetted at Christianity/the Bible but at similar stories as well.
4) People who were non-Christians were more accepting of the beliefs of others (well, 'pagans' were), as they had an ever changing view of religion and gods and their involvement in the world around them. They wouldn't necessarily criticise the Bible (as it was then, which it wasn't really), and would be more likely to think that perhaps the god of these Christians did exist, but was one of many, and not a single god, so they wouldn't say "oh no he doesn't exist", but would (and some did) view him as a god rather than the god, and hence wouldn't attack the writings surrounding him.
5) People did attack monotheism, especially the Jews, and their beliefs, teachings and writings. There is quite a lot which talks about the strange ideas and practices of the Jews and their teachings (which are included in the Bible, obviously), and the Jews themselves were obviously looked down upon by many for their different and strange (to others) beliefs.
6) The early Christians weren't exactly nice people and pretty much set out to eliminate as much of paganism and non-Christian belief as they could, and were fairly successful, so what there may have been which contained a dissenting opinion about Christianity/the bible is also unlikely to have survived due to this as well (ignoring things like the ravages of time, lack of printing press etc), as the Christian leaders wouldn't want people going around spouting off about how the Bible is bunk, and they controlled most of Europe so they could do a pretty good job of basically eradicating all negative viewpoints and all resistance (including groups of Christianity with different views to the mainstream -> look at Christian Schisms through the ages).
I probably didn't explain myself clearly, nor have I covered everything, because there is so much inherently wrong with the foundation of your original question that it would take too long to go through how the world was before Christianity to point out how your question is completely wrong and there was no need to have an anti-bible/anti-christianity thing, because it did kind of exist, and religion then was very different to religion now. (Not an attack on the OP, because not everyone studies this sort of stuff, and so wouldn't necessarily know all the different elements at play).
But that mostly deals with early Christianity, up to maybe 400AD.
Oh, and there was no Bible to attack for a while really, and most people couldn't read, so in the time between Christianity coming to dominance, which kind of coincided with the dark ages, the Bible being formed as it is now, and people being able to read and write and have free time, there were gaps, so by the time there was a fully formed Bible to attack, Christianity had been dominant for centuries, and there wasn't a high literacy rate, and people didn't think to question. (Up to the renaissance etc where people did start to think of new ideas, leading into the 18th centuries and onwards where scientific breakthroughs called even more of the Bible into question)