Does the Bible ever say that our beliefs must be written in the Bible? Must we be able to cite chapter and verse that explicitly says what we believe in order to justify that belief?
If you think so, please cite chapter and verse stating this.
No, I think that would be a misunderstanding, at least to some degree.
Also beliefs are a lot of things, I might believe that taking a cold shower each morning will make me live longer. Im pretty sure that you won't find support for such belief in the bible.
The way I would probably put it, is that one has to believe in the bible as being true and with a higher meaning, not necessarily that it has to be literally true, because some stories are more poetic, some seem more historical etc. but that the messages in it are divine or divinely inspired. I find it difficult to see how the authors of it, would have made people believe it, if that was not their intention. It would fall pretty flat, had they just said that it was some random stories that they made up, because they thought they were cool.
Also at least in my opinion, there is no doubt that the OT was used as law in ancient Israel, I think, there are so many in it, about everything that it would seem unlikely that it weren't. And telling people that it was God that gave them these, would require people to actually believe in it or again I doubt, people would have bought it.
So in that sense, you can say that at least some beliefs would have had to be justified through the bible and those in charge of handling disputes, would probably have read from it in order to show how and in which way people did wrong, what their punishment should be etc.
Personally I think the ancient jews back in those days, would have looked at it much in the same way as we do about the stuff we believe in today and follow. For them this was probably very good and made a lot of sense.