Image noise goes up with higher ISO, fact of life.
Well that's one way of looking at it.
I would prefer to look at it as the noise being reasonably constant (within limits) but the signal being reduced due to the lower light levels available when you have to use higher ISO settings. The end result is a lower signal to noise ratio (SNR) leading to a noisier image. The post-capture ISO amplification applied gets you back to the correct image brightness, but as it boosts signal and noise equally, cannot do anything to correct the already impaired SNR, hence the noise that was there all along becomes relatively more visible.
Impaired SNR is the natural result an under-exposed sensor which of course is what you get if you use an ISO setting higher than base ISO. The exposure meter may say you are not under-exposed, but that is because it is adjusted in advance to allow for the amount of post capture ISO amplification which is going to be applied later. By going above base ISO you are effectively accepting under-exposure at the sensor.
I would therefore say that high ISO noise performance is a somewhat mythical concept, as in reality the performance at any ISO setting is largely dictated by the SNR ratio at base ISO. The better this is to start with, then the better the results as you start losing signal due to low light levels.
Yes, SNR is what matters most to what we see as image noise and you're absolutely right, ISO amplification happens after the sensor is exposed and amplifies the recorded signal at each pixel which includes the noise component... the camera doesn't know what the true signal was and how it was affected by the noise. When shooting in low light the input signal is much lower so the SNR is in turn lower and the image looks noisy.
On the issue of thinking of noise as a constant, that depends what kind of noise we're talking about. There is noise from the electronics (read noise) which is fairly constant and photon noise (aka shot noise) that is not constant. Both contribute to the total noise (random variation) in the recorded signal.
Photon noise scales as the square root of input so the noise goes up with input signal in absolute terms (not constant) but is a smaller percentage of the input (higher SNR). With very low light levels (dark shadows, night photos) photon noise can be significant.
Read noise is fairly constant for a given ISO setting and is in the range of around 2-20 electrons for most CMOS sensors. Again, in dark scenes when the input signal is low (which is when we use higher ISO) the read noise becomes much more significant relative to the input signal (lower SNR) and we can see the noise in the dark areas of the image.
Even when we look at an image we can't "see" or know that a particular pixel (or any area in the image) was supposed to be say 500 photons but was recorded as 495 or 510. What we see is the unnatural variation in adjacent pixels or groups of pixels that wasn't in the true input. So say a group of 20 pixels were supposed to all be 500 photons but they were recorded due to noise anywhere from 470 to 530 we can see that variation and because it's random variation we know intuitively that the image isn't supposed to look that way and it's therefore noise.