In theory, yes. Of course, the same holds true for a lot of things which we currently use clean water for! The water needs of agriculture, toilets, carwashes, and many more could be addressed through so-called graywater (e.g.: pumped lakewater, rooftop rainwater) if we really sat down and wanted to make it happen.

The reason that we don't do these things is rather mundane: it's cheaper and easier to tap into the shared drinking water infrastructure than it is to collect your own water and roll your own silos/filtration tech. That might change as the world changes -- something has to give eventually if we use more groundwater than we replenish, but much like clean drinking water, I don't think it's a problem we should ask individual entities to solve. Governments would generally be much more suited to efficiently collecting drainwater, scrubbing it, distributing it, and mandating usage in wasteful commercial applications.

Dirty water is bad for cooling equipment

The water isn't being consumed. It's going through the same process all the water in the city is going through.

Pulled from the river, cleaned, used for cooling at the data centers, and returned to the river via the waste water system.

The only loss is the energy/resources to treat the water.