Using alpha channel as color amount on material?

I'd like to use an image with either grayscale or alpha channel to control how much of the material color is applied in any given area.

The goal is to create one image and control the color from the material to enable some variation between material without needing a separate image for each variation.

This works with sprites and transparency - a single image can be colored via code.

But I can't get it to work properly with the standard shader materials. I don't want the material to be...

Using alpha channel as color amount on material?