There is one study I'm aware of that attempts to correlate lack of arch support and foot function (including flat foot) in a group of people who do not utilize any arch support.

While not specifically Roman, I'd suggest that this particular population of > 5,000 persons surveyed in 1949 who had never worn shoes, or only thin sandals with no arch support, is projectable to feet from any era, have they not known traditional modern footwear.

Published in the Journal of the National Association of Chiropodists, Samuel B Sulman, Pod.D. set out through China and India to "note the orthopedic condition of those feet in order to compare them with the average of those of us who wear shoes almost all our lives."

Study summary: http://refs.ahcuah.com/papers/shulman.htm

Few functional issues or complaints of pain were noted, and flattened feet as a specific condition was determined at 0.19% or 10 total cases from the entire study.

The conclusion made is that "These figures prove that restrictive footgear, particularly ill-fitting footgear, cause most of the ailments of the human foot."

As such, it appears that flat feet is not caused by lack of arch support. In fact, quite the opposite: "There is no occupation more strenuous for the feet than trotting a rickshaw on hard pavement for many hours each day yet these men do it without pain or pathology."

In regard to understanding flat feet, consider that feet with a flat appearance may be strong muscular feet with a low arch - and that is quite different than what we may see today as a flat foot, same appearance perhaps, but with no strength.

From the same study: "The average height of the longitudinal arches was quite a bit lower than we usually consider average in the United States. But the height of the arch, per se, had no significance when compared with orthopedic malfunction. The lower the arch, the less distance it can fall, the less possible strain upon it and the less possible bony mal-alignment and pain it can cause."

The modern idea of a high arch being optimal is missing half the story. An arch is intended to move fully from its high to low position through gait. Arch support in shoes will prevent the arch from moving through its full range of motion, and in time a loss of foot function results.

If arch support is utilized for children, a lack of intended arch function will result early on, and is typically difficult, but not impossible, to restore later in life. If someone without arch control removes their arch support without doing so incrementally, they run the risk of acquiring flat feet.

In addition, just as high intensity usage in the job of rickshaw driver creates very strong feet ("If anything, their feet were more perfect than the others.") - lack of activity on the feet, and/or ill fitting or supportive shoes can also create very weak feet, or encourage the context for flat foot.

Here are two more sources that may be of interest in regard to the effects of modern footwear: First, published in 1912 "The soldier's foot and the military shoe: a handbook for officers and noncommissioned officers of the line" by Edward Lyman Munson. One section of this book describes how new recruits would arrive in the military with poor foot function due to choosing stylish shoes previously, instead of footwear which would allow proper foot function. And that once established in proper footwear, feet could adapt back to their proper function much more quickly for younger soldiers.

Second, published in 1884 in Popular Science Monthly, an article entitled "Fashion and Deformity in the Feet" also details how modern narrow toed footwear would disrupt normal foot function.

https://books.google.com/books?id=idjUAAAAMAAJ&pg=PA653&lpg=PA653#v=onepage&q&f=false

I add these last two references as part of a final thought: it seems there would be some people in any ancient or Roman times who would have had flat feet, but I suspect they were those who were less active, and/or choosing footwear with a priority of style over function. (and anyone in that position would probably be trying to put anything in their shoes/sandals to help accommodate discomfort, just as that occurs today, be it padding or some other type of support.)

For the rest of the population, working on their feet, using footwear with no support, or going barefoot, I submit that flat feet as we know it today was not common at all, because the lack of support enabled them to properly maintain foot function throughout their life.