The Gaulish tribes, the source of much cavalry, as foederati, had iron horseshoes nailed on by the 1st century CE. So that assumption doesn't make it. These aren't hipposandals tied on.

Secondly, even without them, horses don't have soft feet like dogs or elephants or camels. Ann Hyland, in The Medieval Warhorse (which starts in antiquity) points out that horses training for and competing in 100-mile eventing don't need horseshoes if they live on dry ground. The hooves stay very hard and it takes a lot of wear to outpace hoof growth.

In a wet climate like England, the same horses need shoeing - to keep up the same level of training and eventing. This is probably more mileage at faster speed than most cavalry would do. It doesn't hurt their feet. It just wears the softened hoof down faster.

As for the roads, all the laws limiting equine traffic were to prevent the roads wearing too soon. Iron-tired wheeled traffic was especially abhorred because you don't want ruts where you march.

For cavalry, Hyland is good, whether Equus on horses and other equines in the Roman world, in all functions, or Training the Roman Cavalry, which dissects Arrhian from a horse-trainer's POV.

Any bad effects would be long term, in the sense of horses wearing out in fewer years of service, but they weren't used at the rate of, say, 1820s coach horses. Arrhian is all about refitting battle-shocked remounts for further duty. They took the best care they could, in practicality.

So there is no reason to think the cavalry couldn't use the roads, that the horses couldn't travel at a normal speed, that they would be falling and being injured, or have their hooves destroyed. They would be spared all the dangers of rough ground where weeds hide rocks, thorns, and holes.