There’s quite a lot going on in your question, and I think it’s probably a good idea to unpick some of your assumptions before we get to the business of running as a form of exercise in the ancient world. I’m going to focus on Rome, since that’s what you mention in your original question.

(1) You’re certainly right that ancient people would have walked more than we do in the modern west (but then, so do people in other parts of the modern world!). Most people would probably have been subsistence farmers, and so yes there was more manual labour. However, there were also still plenty of city dwellers for whom work would not have been particularly strenuous – I’m thinking of merchants, traders, craftsmen and such, not to mention enslaved people who were put to work in the homes of their enslavers. At the top of society, the equestrian and senatorial elite focused on business and politics – not exactly manual labour! We know from various texts, especially medical literature, that physical exercise was part of ancient life for at least some people – more on this later.

(2) Roman clothing was not particularly restrictive. The most common garment was probably the tunic, essentially a knee-length t-shirt, under which they likely wore a kind of loin cloth called a subligar. There are some artistic representations of people exercising that show the stripped down to these undergarments (see the previous link), so if a Roman did ever go for a jog, we shouldn’t imagine them doing it in a full toga. For what it’s worth, Greek athletes are often depicted nude. No doubt there were some practical reasons for exercising naked on a hot summer’s day in Greece, but there are also lots of cultural meanings behind Greek nudity, which I don’t really have the space to go into here.

(3) It’s generally quite well known that baths were a central feature of Roman society. Aristocratic houses had their own bathing facilities, and many cities had public complexes that were open to everyone. Did they bathe as regularly as we do? Probably not, but that doesn’t mean we should imagine everyone in the past was always ‘sweaty and gross.’

I think that addresses a few of the base assumptions in your question. There’s no reason that clothing or personal hygiene should have been a reason for someone not to use running as a form of exercise. The jobs that people did might have given them less time or inclination for athletic exercise, but there was a significant subset of the population to whom that did not apply, especially in cities. So, to the main question: did the Romans run for exercise? The short answer is yes, they absolutely did. For one thing, running was a key part of military training. Although a late text (probably 4th century), Vegetius’ De Re Militari makes it clear that running was not only part of the basic training of new recruits, but also the ongoing drills of soldiers at all parts of their careers. Away from the military sphere, civilians also used running as exercise. Many bath complexes (like the Stabian Baths in Pompeii were equipped with a palestra, which was an open space used for running, boxing, wrestling and ball games. Medical writers encouraged such exercises for the promotion of health (see Celsus On Medicine, 1.2.5-6), although they sometimes differed on the benefits of different kinds of activity. In one of his pamphlets (On Exercise with a Small Ball, online here but paywalled), Galen extols the virtues of ball games, claiming that they are far superior to all other forms of exercise, including running, which Galen criticises for causing excessive thinness and also promoting cowardice by encouraging men to run away from their enemies. We can take or leave this medical opinion, but for our purposes it is enough to demonstrate that running was a common thing for Romans to do.

Hope this answers your question. I’m happy to answer any follow-ups you might have!