There's a difference between removing surface stains and whitening. Surface stains, also known as extrinsic stains, come from different sources: coffee, red wine, tobacco, and dark-colored foods and drinks. They live on the enamel layer and can often be removed with toothpaste or surface whitening treatments.
Your type of tooth discoloration may influence the kind of whitening treatment that will be most effective. For some people, a whitening toothpaste might be all they need to get rid of surface stains and brighten their teeth.
That means toothpaste with activated charcoal and other ingredients can likely remove the evidence of your coldbrew habit, but it probably won’t match the results of a professional whitening treatment.