In order to define a god there’s a related term that needs to be looked at and that is human. The meaning of both words may seem obvious: one is divine and the other is, well… human. But this carries more assumptions than meaning and, what’s worst, meaning as defined by Judeo-Christian theology. Which is why the west has so many problems dealing with the Japanese term kami, usually translated as gods or god, but which some prefer not to translate at all and use the word in its original form. The reason is simple: to westerners, the notion of a deity implies something non-human that’s different from or outside the world, but the Japanese recognize kami in the mountains, trees, rocks, rivers, wind, sun, metals, or even human beings. In other words, divinity is an individual quality in equally individual things, part of the natural and human worlds and in everyone of us and everything. It’s not pantheism, but it may be animism. Whereas western Judeo-Christian theology makes human and god two radically opposing categories, Shinto blurs them. And that Japanese perspective seems to be how pre-Christian European and Mediterranean cultures saw the matter, too.
As I see it, a god is simply an otherworldly entity. Not so much in the sense that it’s out of this world, but that its nature goes beyond the mundane. It’s special, standing out from its surroundings and distinguishing itself by its power and ability to cause awe. It doesn’t have to be perfect, morally “good” or even beneficial and it can be anywhere, be it rocks, rivers, trees, graves, activities, human objects, and even deceased people. A god or goddess is something powerful and out of the ordinary, and, therefore, extraordinary: the ancient tree, the towering mountain, the life-giving river, the strange noises in the woods, the amazing skills or intellect, the unexpected moment of luck, the highly bountiful trip or fertile land, the mighty thunder, the ability to influence events from the grave.
This is, at least, what one sees in pre-Christian Roman religion. Gods were not only the greater beings like Jupiter, Mars, and Juno, but also the spirits of the place (genii loci) and one’s ancestors. Deceased humans were actually referred to collectively as Di Manes or the Divine Dead. They too are gods and even greater ones may have started as no more than a local spirit of a hill, mountain, or river mouth: Jupiter at Capitol Hill and Diana at the lake in Aricia or, to use non-Roman examples, Zeus at Mount Olympus and Serapis at the delta of the Nile. Popularity gave Them an upgrade, so to speak, but the reverse can also happen. And then there are also cases of humans made gods after they their death: Quirinus may have been Romulus in life and Antinous ascended to godhood after He drowned in the Nile. To reconnect with the Shinto example above, an also interesting example is that of Sugawara no Michizane, a 9th century Japanese poet and scholar who passed away in disgrace. However, He was given a temple when the unfortunate events that followed His death were accredited to His angry spirit and today He’s popular among Japanese students and scholars.
So there’s no reason why being a polytheist should imply that you neglect the local spirits and your ancestors. That only happens if you’re working based on the Judeo-Christian definition of god, which is very non human and non worldly. Contrary to how pre-Christian deities were seen by at least a large part of Their worshipers. And They also don’t have to be necessarily “good” or morally perfect: the Gods of disease and destruction are hardly friendly, but that doesn’t mean that They won’t befriend humans (occasionally) or, more likely, won’t accept tributes to keep Themselves or others at bay. That’s what apotropaic worship is and it was very common in the ancient world.