TLDR: In Harry Potter, wands are mandated in England (and many other wizarding nations) by their respective governments in order to provide an easy method of enforcing the law.
From some of Rowling's assorted lore, many wizarding societies (like pre-colonial Native Americans or Africans) never used wands. In fact, Uagadou (African Hogwarts) taught students primarily without wands. Since Uagadou was known for having some of the best Transfigurers in the world, we can assume that this had no effect on their power of their magic. We can also see wandless magic in a number of cases where underage witches and wizards pull off impressive magical feats with no wands at all.
However, in the few rare cases when Hogwarts students try wandless magic, it always fails miserably. The school doesn't teach it, not even as an optional course. Why?
Forcing reliance on wands makes it easier for Wizarding governments to keep the peace. Most of their serious (or even minor) crimes are punished by having your wand broken. This provides an easy way for the Ministry and others to prevent repeat offenders, as well as stopping prison escapes.
Imagine how easy it would be to get out of Azkaban if anyone could just cast the Patronus charm with their bare hands. Imagine how absolutely useless "Expelliarmus" would be if your opponent had no weapon to disarm. Imagine how dangerous it would be to have that expelled Hogwarts student go rogue and start blasting up muggles with their bare hands. Hell, even Prior Incantato, the spell that the ministry used to get most of its evidence in magic cases is based on the suspect having a wand. Obviously, the Ministry can't allow any of that to happen, as it would undermine their entire government.
The Ministry of Magic does have the power to impact curriculum at Hogwarts. We see them do this more obviously with Umbridge, but it's shown repeatedly that they set examination standards, skill requirements, etc., and likely have a great deal of influence over textbooks. Wands are required for every Hogwarts student, likely at the Ministry's command.
From what we can tell, there are very few quality wandmakers. Ollivander was the best, and it's mentioned that there are a few others, but it requires a high degree of skill, time, and materials. That means that the Ministry can very easily keep an eye on who gets one, and block any criminals, or the "undesirables" of their choice.
We see mentions of wizards and witches using wands from a long, long time ago, so this likely wasn't some ministry scheme that came out of nowhere. However, once the tradition was in place, the Ministry would have enthusiastically supported it, and made sure that it stayed as the norm.
Additional evidence for this comes from the fact that the two areas that didn't take part in wand usage (Africa and the Native Americans) were the two areas that weren't originally part of the Statute of Secrecy. If you're gonna try to move your society underground, of course controlling who gets magic would be more important.