This is the correct answer. Biological or non-biological powder (not liquid or pods, and not colour care) contains oxygen bleaching agents, and a bleach activator called TAED which speeds up the reaction at lower temperatures. This combination forms peracetic acid during the wash, which itself has cleaning and disinfecting properties. Oxygen bleach is safe to use on colours, but even if you are concerned about darks fading over time, there's no reason why you can't use a liquid for these, and stick to powder for everything else.

Also, one other point: we hear a lot more about mouldy, smelly washing machines nowadays, and there is also far more emphasis on shorter, lower temperature cycles now. The two are very much related. It's not what people usually want to hear, but running longer, hotter cycles at least occasionally will help the machine clean itself through normal use, so it's less likely to turn rank in the first place.

I run my machine no lower than 40°C, using nothing more than regular powdered detergent and occasionally a little splash of softener, and it's still squeaky clean despite our ridiculously hard water. Detergent pods and quick washes are undoubtedly faster and easier to deal with, but there is a price to pay for that convenience.