For instance, when we see religious statues today (such as in Hinduism, neopaganism, and even in Catholicism) it is generally understood that the practitioners of these religions do not literally believe that the "idols" are the deity they're worshipping, but rather, representations of the divine (and not divine itself). Is this true of those polytheists derided by the Bible as 'idol worshippers' as well, or is there evidence that ancient religions really did believe that religious statues were divine themselves?
I've read a fair bit about idols and images in Mesopotamia, so I might be able to offer some insight into what the experts are saying. I will stick to the ancient Near East and not touch on how ideas of idol worship have been used a theological battering ram in later times, although that is also an interesting discussion.
First of all, I hate to do this to you mate, but the answer to your question is a bit of yes and no. Idol worship was most definitely a major element in ancient Near Eastern religions, which as you know is the context in which Judaism was born, and later Christianity and Islam. In Mesopotamia, the link between god and image may well go back to the earliest religions in the region. At the same time, you are right on the money by asking whether idols weren't in fact understood as representations of deities.
The people of the ancient Near East did make a distinction between the deity and the idol which represented the deity, and the deity was not constrained to a wooden statue that gathered dust in the heart of the temple. However the idol was definitely divine in itself, and although the exact relationship between deity and image is debated, the interpretation that is the most convincing to me is that part of the deity resided within the idol. This part of the deity was very central for the people who worshiped it, by being an aspect of the unfathomable supernatural made perceivable to human senses.
The ancient Near Eastern city had a patron god that inhabited the city through its idol. The idol served as a concrete way for the priests and sometimes kings to interact with the deity. The idol was dressed, fed and given offerings, and in turn it could answer prayers. As long as the god was respected and taken care of, it ensured the city protection and well being.
The divine nature of the idols is especially made clear to us in their creation/birth, and in their destruction/death. After a new idol had been made, the Mîs-pî ritual, or the "washing of the mouth" was performed to consecrate the idol and remove any trace of human craftsmanship. Through the ritual the idol came alive, and could sense the world around it. Now, if the idol was destroyed, inscriptions tell us that the entire city would grieve it, much like when a beloved community member died. From what I have been able to gather, the deity generally did not die alongside its idol, although in some cases it did. This was the fate of the goddess Ba'u of Lagash for instance, when Elamites tore down her temple. In other cases the deity was wounded when its idol was destroyed, or the deity could be banished to a sort of limbo state until the idol was repaired or recreated, which was no simple task. Although a temporary symbol or idol could be used in its stead.
It should be noted that a city whose patron god was destroyed or abducted by an enemy was considered extremely vulnerable to all manner of attacks and misfortune, and essentially already doomed. This made the idols prime targets in war, and we have historical and archaeological evidence of idols being smashed, vandalized, disgraced, abducted, and in some cases even refashioned by the conqueror. This was a cunning way of bringing the defeated city and cult into the sphere of the conquering empire.
To summarize, the consensus among the wise appears to be that it's a bit of yes/no situation, that may come down to your definition of idol worship. Idols were divine and worshiped in some sense, but they also weren't literally synonymous with the deity. I hope you found this interesting!
Edit: Here's some quality works if you want to learn more
Curtis, E. M. (1990). Images in Mesopotamia and the Bible: A Comparative Study. In W. Hallo, B. Jones and G. Mattingly (eds): Scripture in Context III: The Bible in the Light of Cuneiform Literature. Ancient Near Eastern Texts and Studies, 8. Lewistown: Edwin Mellen.
Dick, M. B. (ed.) (1999). Born in Heaven, Made on Earth. The Making of the Cult Image in Ancient Near East. Winona Lake, Indiana: Eisenbrauns.
Hallo, W. W. (1983). Cult Statue and Divine Image: A Preliminary Study. In W. W. Hallo, J. C. Moyer, and L. G. Perdue (eds): Scripture in Context 2: More Essays on the Comparative Method. Winona Lake: Eisenbrauns: 4–5.
May, N. N. (ed.) (2012). Iconoclasm and Text Destruction in the Ancient Near East and Beyond. The Oriental Institute of the University of Chicago. Oriental Institute Seminars 8. Chicago: The University of Chicago.
Ornan, T. (2005). The triumph of the symbol: Pictorial representation of deities in Mesopotamia and the biblical image ban. Ribourg /Göttingen: Academic Press/Vandenhoeck Ruprecht.