You got the cause right: reducing exposure is what causes noise.
ISO correlates with noise and it also correlates with the brightness of the camera's output image (JPEG). As you raise ISO the output image gets brighter. And both these correlations between ISO and the camera's output image (JPEG) are strong. There's little you can do to break them.
But as I previously noted the correlation between ISO and noise in raw files is much weaker -- same goes for the correlation between ISO and image brightness. You don't absolutely have to change exposure with an ISO change.
Raising ISO without lowering exposure won't over brighten the output image if it's too dark to begin with. This is what I caught my students doing. Misinformed as they were about ISO causing noise, and aware that they couldn't increase exposure because of motion blur they would raise ISO to what they thought was their noise tolerance level and then leave the camera output image too dark -- I noted my student said he planned to fix it in PS. They thought they could game the system by not raising the ISO even higher and so have less noise in the photo they adjusted in PS.
They were wrong because they were misinformed and confused about ISO causing noise.
Because the correlation between ISO and noise/image brightness is weak with raw files it's easy to do this: Took two photos for you.
One at ISO 200 (base ISO of my camera) and the other at ISO 3200 -- both the same exposure: 1/15 sec, f/5.6. In both I processed the raw files.
A) ISO 200:
ISO-200-noise.jpg
B) ISO 3200:
ISO-3200-noise.jpg
Because the exposures are the same the shot noise level should be the same in both photos and it is. The ISO 3200 photo has very slightly less noise because of some read noise suppression -- camera is doing that when the ISO is raised. Here's a 100% peak of the two side by side:
View attachment 269282
If raising ISO really is the cause of noise why does the ISO 200 photo have more noise than the ISO 3200 photo? If ISO amplification is causing noise then why does the ISO 200 photo (less amplification) have more noise than the ISO 3200 photo? The cause of noise is reduced exposure.