I'm wondering how the color ramp affects the noise texture for the texture brush exactly? It's important that the black marker on the colour ramp's left end has alpha set to 0 because otherwise you get colour dots on a black background as you also get with "Colour Ramp" unchecked.
spikeyxxx It was unclear to me why I need a color ramp for this to work since the noise pattern with it's greyscale value (and black and white as the two extremes) could already define the brush colour opacity.
In the meantime, I've tested the behaviour since I couldn't find any description in the Blender manual (right click on the colour ramp check box and selecting "Online Manual" only directs you to an overview page for the different procedural texture types):
The colour of your brush acts as a filter for the colour ramp gradient so that you simply need to set your brush colour to white in order to get the exact colours from the colour ramp since white contains all three primary colours. Therefore, using black as brush colour will result in a completely black stroke no matter which colour you choose for your colour ramp.
Not using a Colorramp would mean that black from the noise would be black (disregarding the Brightness here).
This would be the same as using a Colorramp with the black having an Alpha of 1. (Like your standard Shader Editor Colorramp.)
But, because this Colorramp has its black with an Alpha of 0, this means that darker values from the Noise Texture are more transparent than light values.
The color of your Brush comes after the colorramp! Here is an example of painting without (left side), compared to with colorramp;
the top is with a red Brush and the bottom with a black one:
So the order is: Noise > Colorramp > Brush.
Hope this helps.