Opinion on how to finish quarter round
I am one of those yahoos who prefer the shoe to match the floor color, although IMHO it is a matter of personal opinion. In the high end homes I help oversee being built in the Washington DC area, the color was determined by the architect or interior designer, of which many of these guys were highly regarded in their field and there was choices made of both types of finish. Again personal opinion...

In the case for the question, I would stick with the painted option, even though I prefer the stain. The stain would look different than the original, unless the floors will be refinished, then the floor finishers stain it along with the floor. The painted version will be easier to match up, and it may not hurt to "face off" the base with another coat to help hide the joint where the shoe and base meet. That way if the shoe is caulked in place, it has been done before, the caulk will receive a coat of paint to protect it.