You're close to hitting the nail on the head. In the late 19th century, prior to the advent of air conditioning or even electric fans, one method of resisting the summer heat was to dress in white cotton or linen - if you were affluent enough to not be working somewhere your clothes could get dirty, that is. At this point in time, doing laundry was an incredibly onerous business. Most darker/heavier materials could stand to pick up a little dirt and be brushed off and hung up for wearing again - sweat not being too much of a problem due to the layers of underclothing that would soak it up instead - but anything that happened to white fabric would have to be properly washed out, and so white clothing was worn sparingly. So the original version of the "rule" to only wear white in the summer was less a prescription than a description of ordinary habits of affluent people.

Around the turn of the century, white started coming into fashion in a more general way: first with the stereotypical 1890s white shirtwaist, then with white gowns. With the rise of the garment industry, these were available at more moderate prices and were worn by larger numbers of women than had previously worn white during the hottest months of the summer, especially in the country. The shirtwaist, paired with a darker skirt, even became the female office worker's uniform. As is usual when an item or manner of dress that used to signify class becomes widespread, many were uncomfortable with the change. For example, a writer for The Young Woman's Journal complained in 1908 of how ubiquitous white dresses were compared to the past. The Dry Goods Reporter noted in 1905 that "it will make it distasteful to well dressed people if the wearing of white canvas shoes is persisted in where it should not be seen - on city streets."

Unfortunately, I don't know of any scholarly work that traces back the specific Memorial Day/Easter/May 24th-Labor Day wording of the rule, and after 1920 or so the primary sources become difficult to track. It doesn't seem to have been explicitly set in print until the 1970s, being rather the kind of thing that mothers passed down verbally to their daughters, like most other class shibboleths. It also seems to have been mostly about the white shoes rather than white anything else in particular at that point.

I heard that this fashion rule was created before there were washing machines and it was much easier to wash clothes in the summer. It was much more difficult to wash clothes / hang clothes out to dry when it was freezing outside.