The noise is not created by the ISO setting. Neither does it lower the Signal to Noise ratio. So not really chicken and egg.That is, it's actually ISO setting, when applied to the resulting image, creates visible noise. But we used the higher ISO because the image was too dark (the information was in the raw shadows with low SNR). So it's like chicken and egg problem.
Increasing the brightness of an image will however make noise more apparent in regions with poor SnR, as you say. Raising this brightness in post is not superior to doing it through ISO though if noise is the primary concern.
I'll post some pictures I took yesterday later.