First of all it is important to distinguish what time period we are talking about, roman footwear changed dramatically and the footwear of the first century looks absolutely nothing like the footwear of the fourth century. So I will assume you are referring to early imperial period of the mid-late first century, since that is what most people think of when they think of the roman army.

We call roman footwear "sandals" because of their physical appearance being similar to modern sandals. However, they were not "sandals" as the definition of a sandal is "a type of light shoe consisting of a sole held to the foot with a series of straps or cords." Romans DID have sandals, called Solea, [modern reconstruction of a roman solea] (https://imgur.com/a/AVtIWbJ)

For military footwear, they had two types: a sandal like shoe called "caliga (plural caligae)" and a boot like shoe called "calceus (plural calcei)" both are a made from a thick leather sole, with a thin leather upper, and both usually had metal hobnails on the bottom.

Here is a modern reconstruction of a [Mainz style roman Caliga] (https://imgur.com/a/prKLUbZ) now compare that to a reconstruction of roman [calceus from Koln] (https://imgur.com/a/rGSBeXA)

You will notice that both shoes look very similar, the only difference being that the slits cut into the caligae extend to the sole, while the ones in the calcei are only on the top of the shoe and it has an enclosed toe. Essentially the caligae is really just a boot with lots of slashes cut into it. This becomes more apparent when you look at a [pattern of the leather spread out] (https://imgur.com/a/itqXDc7)

This distinction becomes even more blurred because many of the calcei began to have very elaborate examples of patterns cut into them as well, with a netlike pattern of circles and squares cut into them. [example 1] (https://imgur.com/a/Mqua4is) on the bottom right, [example 2] (https://imgur.com/a/l9IKVw4) on the top left, [example 3] (https://imgur.com/a/swL0WkQ)

This open style of shoe allowed for a comfortable fit that allowed the foot to breath, and water to evaporate and drain out. This is important when wearing leather shoes without modern waterproofing so your foot doesnt just sit in a puddle of water and can lead to blisters or even worse foot problems. Especially when going on long marches, as the Roman military often was expected to march up to 25 miles in a single day

As to your point about protection, it is important to remember these are made out of thin vegetable tanned leather, 3-4 oz leather. If you dropped your knife and it hit your foot, that leather would probably stop you from getting a small cut. But if an enemy is thrusting a spear towards your foot with their full force? that 3-4 oz of leather would do absolutely nothing to stop the spearhead.

So since you gain much by allowing your foot to drain properly, and lose almost nothing realistically in protection during combat, this seems like a good tradeoff particularly since soldiers would be marching waaaaaaay more often than they would be in combat.

Edit* for more information about Roman period footwear, I reccomend looking into Carol van driel murray, she is the foremost expert on roman leatherwork, and she has written dozens of papers, articles, and books on roman footwear, both military and domestic

Edit 2* since so many people have asked about socks I will add a little bit about them. We know roman soldiers wore socks, as there are several wood tablets from the Fort at Vindolanda in Britain in which roman auxiliary soldiers mention socks. One in which a soldier writes home to his family asking for them to send socks and other items of clothing, and another record of several items sent in a shipment to the fort. Additionally a roman hobnail found in yorkshire was found that still had some natural materials stuck to the nail bit of the hobnail, including some leather from the shoe itself, as well as some natural fibers, which were likely from a sock.

Roman socks came in two types. The first was a fabric sock, cut from linen or wool fabric, and stitched together into a short sock example here

The second type is a knit sock made in a stich pattern called "naalbinding" which is a bit like crocheting. We have far more examples of these, as hundreds have survived from Egypt and the levant. They were often made in brightly colored contrasting wool, and were often made like mittens, with separate spots for different toes, likely to allow the wearing of a thonged sandal, where the thong goes between your big toe and the rest of your toes. example 1 example 2 example 3

I just did a presentation on Footwear for my reenactment group, so I feel like I might be able to weigh in on this:

So Roman (military) footwear is quite a broad topic because as you can imagine for a civilization that lasted for many centuries, things changed a lot, and much like nowadays, different styles of footwear were in use at the same time, often for different purposes. The stereotypical (even back in Roman times) Roman military boot (it was open, but it came up round the ankles) is the caligae. This is the famous type of boot which looks like a nailed sole with lots of strapping wrapping round the foot joining at the top of the foot. The earliest example I know of caligae found archaeologically that has been found was on a Roman shipwreck at Commachio, dated to around 20-10 BC, while the latest dated example of a caliga found is one found in the auxilia (non citizen soldier) fort of Castleford in northern England, dated to between 70 and 86 AD. So in other words, going by archaeology (and the sample size of roman footwear found archaeologically is fairly large so this is probably a reasonably good indicator) It seems that the Caligae were only in use for around 100 years or so. They do keep cropping up in sculpture of soldiers for another 100 years (See the arch of Septimius Severus for example), but it seems likely this is more of a case of artists showing what they think soldiers should wear rather than what they did.

As to the question of how practical these open boots are for soldiery? Well the answer would of course vary, but it seems they did the job! One thing to bear in mind, if that protection from weapons and the like is not necessarily going to be a high priority. You mention in your opening post, about construction sites. Well thing the legions did a lot of was construction! Almost certainly more than fighting, which is generally not a particularly common event, then or now. And as for construction, bear in mind that even with enclosed boots, we are only talking about a layer of goats skin covering the foot, (steel toe caps don’t exist in this period). So regardless of what footwear you are wearing, foot protection from violence is going to be limited. That said Roman footwear could certainly be hazardous to wear in combat! During the fight over the Temple in the Siege of Jerusalem, Josephus recalls one centurion slipping on the flagstones of the courtyard while leading a charge due to his nailed boots, resulting in him being surrounded and killed by rebels. They do provide excellent grip on unpaved surfaces however, and bear in mind that for an army on the frontier, even in a Roman world, much of the world still only had dirt tracks at best, ideal for nailed soles.

However the nailed soles are pretty much an aspect of Roman, not just military footwear. At Vindolanda, (a fort on the Stanegate frontier in Northern England) where over 5000 Roman shoes have been excavated to date (and they are still finding more), the smallest hobnailed shoe found is thought to have belonged to a 12 month old child (the largest I have heard about conversely is apparently a size 15!).

Vindolanda is great because it provides a fantastic cross section of roman society at that place through its footwear, showing what men women and children of different societal and temporal strata wore over the lifespan of the fort (established about 85 AD, abandoned in the 4th century). Interestingly, among the fairly extensive footwear collection from the earliest period of the forts history (85 to 100ish AD), none of the military footwear found has been caligae, with nailed enclosed boots (often with elaborate fishnet style cutouts) being most common.

That said don’t be under the impression that Romans only wore nailed boots everywhere. One of the most common styles of shoe in the ancient world, the carbatina is essentially a piece of leather sewn up at the back of the foot, and laced up around the foot with a drawsting. At a well in Welzheim where 160 shoes were found, Carbatinae made up 40% of the shoes found, and seem to have been used as an indoor shoe, on hard surfaces, which as they were unnailed, would have been far more comfortable to wear. As a note, carbatina style shoes seem to have been the dominant style among the non Roman European peoples from what we can tell, with nailed footwear being a distinctly Roman fashion.

Though it isn’t directly related to your question, If you will permit, I’d like to add that much of the Roman footwear found is rather more complicated than just a flat sole and an elaborate upper: There have been many example of slippers found in the graves of Roman women which feature cork inner soles (sandwiched between leather outer and in soles), which presumably would have provided a level of cushioning similar to modern foam soles. Interestingly these have only been found in female contexts so far as I know. Additionally, and more universally, many of the shoes found feature the addition of laminae. These are strips of leather, again found sandwiched between the inner and outer soles and fixed in place with thonging, used to give the inside of the shoe complex shapes, allowing the shoe to fit the foot better than just a simple flat surface.

Sources

Footwear in the North-Western Provinces of the Roman Empire, Carl Van Driel Murray

Pannonian Burials: Cork-Soled Slippers as Grave Goods, Judit Pasztokai Szeoke

Douglas,

Charlotte R. (2015) A comparative study of Roman-period leather from northern Britain. MPhil(R) thesis.

Vindolanda Roman Shoe Webinar (Youtube Video by Vindolanda Trust, and presented by Dr. Beth Greene) https://www.youtube.com/watch?v=snf-16yD3JI&t=3051s&ab_channel=VindolandaTrust