Noise is inherent to high ISO, but there are things you do which can hurt it unnecessarily...and underexposure with 'pushing' during post processing definitely hurts the issue of trying to achieve low noise in high ISO shooting!
I honestly think that the average digital photographer today is overly sensitized to 'noise' (commonly mistakenly called 'grain'). Just as film resulted in increase of grain size at higher ISO, digital noise increases with higher ISO. And the byproduct is not only more 'grain' but a loss of DR. Yes, it is OK to want less of the undesirable, but it is also somewhat unrealistic to expect 'no noise' in our images when high ISO is used. It simply comes with the territory!
I have an area in my house in which I can fairly well recreate EV-1 lighting at will, so I set up a shot of a gray target and a Colorchecker. Here is a crop taken from the high ISO shot, using noise reduction within Lightroom...I think it is really low noise

Then it dawned on me one day, that when I look at a lot of high ISO shots that are posted in the high-ISO picture examples threads, one thing that often strikes me is that the photos are 'unnaturallistically bright'...the original scene did NOT LOOK THAT WAY...nightime scenes look more like daylight!
So maybe the right way do shoot the picture is indeed to give it plenty of exposure to increase the signal:noise yet then reduce the brightness in postprocessing to mimic WHAT YOU SAW with your eyes, and not make it look like daylight! When illumination levels get really dim, cones stop working as well and color saturation drops, and more of our rods, the brightness receptors, in the eyes take over...so a photo with those characteristics more closely mimics our vision (and not the electronic sensors's version of reality).
On the left is as-shot by a Canon 7DII, and you can see that I had to use 1/10 f/2.8 and ISO 12800 to get this shot.
Oon the right is a virtual copy in Lightroom, with the image made to mimic
what I could really see of the targets with my own eyes in that dim light.
What I had to do in Lightroom settings was to reduce Exposure by -3EV, cut color Saturation, and boost Blacks and Shadows...as light dims, the cones in our eyes do not work as well in detecting colors, and the rods come more into play to simply detect shapes and motion.
So just maybe our attempts to get low light shots to be bright is UNNATURAL and not like our eyes see and as our brains remember the scene. So if we up exposure in camera AND drop it in postprocessing, we can reduce digital noise yet increase the reality of the rendered capture! Just a thought for folks to chew on.