Beliefs don’t matter in most ancient polytheistic religions. Actions do. We are our deeds. Blessings flow where offerings go.
Ritual matters. Offerings matter. Beliefs are intangible and changeable based on human whims. Doing a ritual in your head is not an adequate substitute for being able to do a ritual physically. If you can’t do ritual due to travel or other reasons, then either erect a travel shrine and do some sort of minimal devotions or say a prayer and offer some food. At least it is doing something (and not just in your head).
Also, Holy Powers exist. Whether or not I believe in Them doesn’t stop Them from existing. Does person X stop existing because I think it? Do the people that I don’t know (which is most of the people on the planet) stop existing because I don’t know them or haven’t met them? If you haven’t met any deities, that doesn’t stop Them from existing either. That just means you haven’t met any. The Holy Powers do not exist based on us, we exist because of the Them. (And also Holy Powers by their very nature are incorporeal. Saying that They can’t exist because we can’t see Them physically is like saying a cat should be a dog because you want it to bark.)
But for the ancients, the existence of deities was a given. This was already built as a baseline for society and religion. “Of course they exist. Offering to Them is proper behavior to be a good citizen/tribe member/family member,” an ancient person may say. “Do you breathe? Do you feel the sun on your back? Can you read, learn or write? Speak? Sing? Heal others with herbs or medicine? Are you proficient in science or math? Do you not see all Their manifestations around you?” One of the ways in which the Holy Powers can manifest Themselves is through us. We are Their Hands and Eyes in the world; Their Work in the world is done through us.
Because I live in a society where deities existing (except one) isn’t the default setting, I had a moment where I realized this was real and not just in my head (I think it helped me since I was not raised in a religion that included the idea that polytheism was possible). I felt this presence in front of my face. I was awake. It was a cat/lioness face and I heard a gruff voice say,”Sekhmet. Offer Me milk.” I heard this as clearly as if another physical person had been in my room. I was alone. After my initial shock, I thought, this is real. This was Sekhmet and those who know Her know She’s about as subtle as a fiery inferno. Which is to say, She’s not subtle at all.
So do I believe in Them? Yes. But They’d exist regardless of my belief. Your actions and offerings bring Their blessings not your belief.
Belief flows from practice. Practice creates and informs belief. Gods may show up first, but then your practice flows from that experience. You offer and They respond. They show up because They have something for you to do, not something for you to believe.