The fixation on whiteness is a cultural and arguably psychological phenomenon as a proxy for cleanliness and not something that should is survived germ theory.

For example, while using white makes sense for being able to identify literal dirt, it doesn't really make sense for modern hospitals given what we know about microbrial life. As noted elsewhere, tarnished copper looks like shit but is actually pretty great as a material.

This is from an article on the emergence of white coats for medical staff:

It appears, then, that rather than being signals of aseptic surgery based on scientific bacteriological research, the white aprons, gowns, jackets and coats were more likely to be the sign of the new “trade mark” aspiration of the expanding middle classes: bodily cleanliness and purity, made more accessible by the industrial production of new fabrics and the means of washing them. Advertisements on billboards and in the rapidly expanding popular press for mass-produced goods such as the high-profile Pears’ soap and the Sunlight washing products from Lever Brothers used key words and phrases such as “purity,” “health” (Stubley 2012, 129), and the “virtues of cleanliness”—the latter made evident in one of the advertisements depicting a naval officer in his tropical whites about to assume “the white man’s burden” of “brightening the dark corners of the earth” by introducing some fortunate “natives” to Pears’ soap (Figure 3).

In a similar way, the fashionable British surgeon and antiseptic “denier” Lawson Tait attracted his middle class gynecological patients not with scientific claims, but rather with visual reassurances of cleanliness and with “rhetorical and aesthetic vehicles of persuasion” (Greenwood 1998, 103)

Tait insisted that hospitals of the period should be “meticulously clean” and models of “domestic hygiene and comfort.” He thought that dirt was “inconsistent with good health and good living” (Greenwood 1998, 122) and that “advance within the art [of surgery] entailed the need for scrupulous cleanliness” (Greenwood 1998, 124).

Understanding the nineteenth century as the great century of linen (Corbin 1995, 13–38) allows the historian of surgery to see through fresh eyes the meaning and significance of the shift from the black frock coat to the white coat. It was not asepsis but pristine whiteness as the sign of cleanliness that led the way. Industrial revolution and chemical discovery made for whiter than white. By donning white, surgeons stepped into a new ideological system.

https://www.tandfonline.com/doi/full/10.1080/1362704X.2015.1077653