Nothing proves that spikes per se are causative either. But there sure is a lot of correlational data for both.

Well it is a cute idea to suggest that we are as unconfident about spikes as we are the LFP. But I would like to hear your mechanistic explanation for LFP entrainment. Keep in mind the LFP is roughly +/- 2mV in amplitude (broadband amplitude), and the resting state of the neuron is around -70 mV.