I have no idea how to interpret “improve our conscious contact with God” any other way.
Then you're not experiencing any empathy for them. You're not actively putting yourself in their perspective, their world. You're accepting what they say, not extrapolating from that to understand what they think.
Religious people generally don't hear voices in their head. We know God doesn't talk to them. They know God doesn't talk to them. They might believe in signs or whatever, but they don't hear a voice when they pray, and they certainly don't expect to.
From the outside perspective of an athiest, you should be able to see that all they're really doing is using their imagination to simulate a being greater than themselves and then asking "what would that being want for my life?"
This is not very functionally different from asking ourselves "if I was a better person, what would I want for my life?"
The theistic process could be corrupted by malformed ideas about the things a deity would want, sure. But the athiestic process could also be corrupted by malformed ideas about the things a good person would want.
I'm not questioning the value of non-electoral political action. That is just as - if not more - important. Get involved. Use your voice. Donate. Rally. Please.
I am only challenging this naive idea that "not voting" = "protesting". You cannot protest by staying home. You cannot protest by sitting out. Not voting isn't action, it's inaction and no revolution will ever, ever start with inaction.