I'm going to make separate posts. First, let me give my definition for Paganism and how its core ideas differ from Abrahamic religions (particularly Christianity).
"Pagan" is a Christian word to refer to the "other". From wikipedia: "Paganism (from classical Latin pāgānus "rural", "rustic", later "civilian") is a term first used in the fourth century by early Christians for people in the Roman Empire who practiced polytheism or ethnic religions other than Judaism." Hindus are pagan, Shinto is pagan, Buddhists are pagan, Wicca is pagan, the Church of the Flying Spaghetti Monster is pagan. You make up a Pony-religion, that's pagan by this definition.
When we (including you) say pagan though, typically we're referring to people who follow ancient or reconstructionist European religions. This is the definition I will use, because typically Hindus and Shintoists don't call themselves pagan. Hindu, Shinto, and Buddhism are very pagan-adjacent and will come up often in this discussion since they are used in the reconstruction process.
Paganism is, and has always been, primarily orthropraxic ("right practice") vs Abrahamic orthodoxic ("right scripture"). This is a critical difference of mentality. Christianity does have some orthopraxis, but that is not the core of the religion.
Let's take Shinto as an example since it is orthopraxic (all indigenous religions are) and currently practiced by a heavily Westernized civilization. The rituals are done to create fertility, honor the dead, purify a space, marry two lovers. The average person participates in the ritual because It's Just What You Do. In modernity many perform this rituals even though they'll claim to be atheist - they do it "just to be safe". Many Japanese businesses keep temples to Inari (fertility, wealth). Next post I'll dive more into Jung and the psychology of these rituals and the purpose they serve for society - here I'm trying to set up the basis for orthopraxis. Essentially, you do the ritual because it's the spiritually right thing to do, and the priests craft these rituals to achieve a purpose (or at least the ancient ones did, it's unlikely modern ones know what they're doing and just keep the torch alive). Christianity has these too (Baptism, Eucharist) but it is not the focus of the religion. It is most likely that these rituals were incorporated into a totally orthodoxic religion idea to ease the transition from Pagan to Christian. Pagan religions have Ritual at the core.
Here's a short >reddit thread answering how the Romans/Greeks viewed their gods to reinforce this orthopraxic idea. There are two good comments here that sum it up well.
I think this mentality difference is the reason for your demands of "proof", and the Christian's need to argue for proof - you come from an orthodoxic mentality. A text is the foundation for your belief and must be "real" to believe in the religion - Christianity could not exist without the Bible. But a pagan needs no texts, no scripture. The Vedas, Eddas, Mabinogion, could all be torched tomorrow and the essence of paganism would still remain.
"What about myth?"
Myth is in a circular relationship with Ritual. Pagan Rituals and Myths change to suit the people, the climate, and the culture. Orthodoxic religions adhere to Scripture, which rarely changes.
This is the reason that Indo-European studies are interesting. We can see the common myth of the Thunder god slaying the Serpent, the divine Father, the Dawn goddess, the interweaving of Fate, and other concepts as they spread across Europe. We can see how they changed to suit the people.
Let's take Aurelius:
Live a good life. If there are gods and they are just, then they will not care how devout you have been, but will welcome you based on the virtues you have lived by. If there are gods, but unjust, then you should not want to worship them. If there are no gods, then you will be gone, but will have lived a noble life that will live on in the memories of your loved ones.
This is clearly an evolution of orthopraxic philosophy. If paganism had continued in Europe, I think we'd have seen some truly great minds expand on this. But it really highlights the idea "proof is irrelevant - do the right thing because it is the right thing".
Okay, that's great to a layman, but why would an intellectual believe in it? And here we get into the core of things - Myth and Paganism is layered. You can believe that Thor genuinely brings rain to fertilize crops and sacrifices please him, or, you can see it that Thor is a grouping of all of the concepts that represent Storms to a Viking-era Norseman - and the sacrifices are a ritual that still serves an important role for society. To a modern pagan, "What does it mean that the Germanics ascended Wodanaz, a previously-unknown wise man, to usurp Tyr, their Zeus?" is a far more relevant question than "is any of this real". It's about what the rituals and myth mean and how they affect your life. I'll expand more on this.
Some more things about paganism. There are three major groups: Wicca, Asabros, and everyone else.
Wicca is based on outdated scholarship and seems to be men's way of convincing women to dance around naked by a fire by convincing them there's a unified feminist moon goddess. They are the most plentiful pagans, they are pretty much entirely women who do stuff with crystals.
Asabros are those who are, as one user with a Japanese name I'm too lazy to tag put it, LARPing atheists. They go around with vegvisir tattoos and talk about how badass vikings are and yell HAIL ODIN a lot. They're like the masculine response to Wicca and equally stupid.
I personally don't consider either of these groups real pagans. I don't think anyone else does either.
"Everyone else" could also be called Reconstructionists and they're who I'm talking about here. We spend an inordinate amount of time looking at archeological publishings in foreign languages to piece together what the pre-Christian Europeans believed. We read about Hinduism and Buddhism and Shintoism to see how our ancestors might practice today. We're a bunch of goddamn losers. But this is also something that informs the philosophy - this is a constantly moving target. At any time an archeological dig could bring something new to the religion - but the orthopaxic nature (several Shinto shrines practice different worship of Kami than neighbors) means that there will never be full contradictions. The malleable nature of modern and historical Paganism tells us we can and should fill in the gaps with our own Gnosis. But the knowledge of what came before guides and gives structure to this praxis so it is not completely unmoored from the psychological truths of ritual.
Next I'm going to sit and watch the videos you posted in the other thread and see how they relate to polytheism, and write more directed at your actual question next.