Why do domesticated horses wear shoes and wild horses don't?

because the wild horses don't have people to put shoes on them.

A long time a go, in medieval times, the King decided he would be safer on a hill. But on a hill, the horses didn't have land to roam in pastures, so they had to live in small pens or stalls, standing for hours in their own waste, which had ammonia in it. The ammonia weakened the hooves, so when the horses were ridden on the cobblestone roads, their hooves shattered. So, instead of actually fixing the problem and turning the horses back out to pasture as they should be, they just put metal on the horses feet, and actually made it worse, although they didn't know it was worse because it seemed better since the horses feet no longer shattered. So that is how horses started wearing shoes. But now, many people have gone back to the way nature intended and are leaving their horses barefoot, turned out 24/7, without shoes, clipping, stalls or blankets, and this is the healthiest way if you are willing to make the change and research everything about it.

The reason wild horses do not need shoes is because their hooves have not been weakened by ammonia and holes in their feet caused by shoes. Also, they run on hard ground (despite what many people think, they do not run on lush green pastures, but rather rocks rough enough to be used as a rasp, which is how their hooves stay trimmed at a healthy length) and their feet adapt to running on this hard ground and grow to be as strong as the rocks they run on.

Domestic horses can have feet as hard as the mustangs if you take the time to gradually introduce them to hard ground, like asphalt, and they too will not need shoes.

So, shoes are an uneducated quick, easy cover up for messed up weak feet, which domestic horses have because they have not been cared for the way nature intended.