I'm wondering if making people spend and transact with me has any consensus power in one shape or an other.

Would it give my node more "value" if it was to verify 10 transactions per day?

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Yes, I think so. You produce a good that is valuable to others, and you wield your Bitcoin node to project power on your clients, saying, "I will send you this valuable good if you send me Bitcoin that my node recognizes as valid.

By doing that, you incentivize your clients to use nodes that agree on the same rules as your node. You're projecting power over the version of Bitcoin OTHER nodes in the system are running. Your power over consensus does in fact increase as your clientele increases, because you influence more nodes in the network to be in consensus with yours.