As things stand today in Stack Exchange, for audio screen reader users they never hear about the coloured (colored) icon because it is tagged aria-hidden="true".
However, not all visually deficient users use screen readers (i.e. 8% of males have red/green blindness). Also bear in mind that this is not exclusively an accessibility issue - absolutely everyone can be temporarily visually impared, e.g. try using a laptop, mobile device or TV in bright sunlight.
The WCAG guidelines play their part by specifing rules, e.g minimum contrast ratios, or best practice like don't use colour (color) alone to communicate meaning. WCAG also provides suggestions on how to pass each guideline based on your technology.
However WCAG can only take you so far in ensuring your solution can be used by the broadest range of human limitation as possible, whether these are permament conditions from birth or permament conditions due to accidents or temporary conditions due to breaking your arm or bright sunlight on your display device.
If a designer is not thinking about the range of human limitations in design, then don't expect your developers to turn your design into an accessible solution.
For badges you can vary the shape of the icon and/or you can vary the saturation of the colours so that Gold has the highest colour saturation and Bronze has the least colour saturation (remembering that Bronze still needs to meet the minimim contrast ratio defined in WCAG) - the quick test is to print the solution or design in greyscale not colour and check you can clearly see distinct shades of grey for Bronze, Silver and Gold, where Bronze has the lighest grey and gold has the darkest grey.