You are both right. Leaving aside edge effects, the sum total of photons hitting 4 1/4-sized pixels (sensels) will have the same Poisson distribution as 1 full-sized pixel but the smaller pixels will lead to more noticeable granular noise. It's a trade-off between detail and noise. But, it's easier to extract detail from a noisy background than it is from one where the detail has been smoothed.Yep. But all four smaller sensels do not necessarily all receive the same number of photons. Poisson distribution and all, not to mention edges between objects in the scene with different brightnesses.
It leads us on to what is the optimal size of pixels for the current sensors. I slightly prefer the heavily cropped files I get from my 50 Mpx 5DSR (20 Mpx APS-C equivalent) to those from my 32 MPx 90D, although when not heavily cropped both are pretty indistinguishable.