The real limiter on this, in D&D, is numbers. You have to be a fifth level wizard to animate dead, you only get a couple per casting, and your control only lasts so long. This works fine for adventuring.
It is not so good when considering building a large body of undead to do work. Nor does it work if you only want a couple of house servants - you still need a living person there to control the undead. In general, it looks like the undead servants need more, and more highly trained, oversight than human slaves do. Does the guy who can throw fireballs (or cast whatever other impressive 3rd level spells) really want to stand there overseeing grunt labor every day?
Note also that if your overseer doesn't case Animate Dead every day, you have uncontrolled undead walking around.
So, this all isn't an argument against the character concept. It is an argument that the character's desired scheme... probably won't work. You need too many 5th level wizards with nothing better to do with their time than oversee mindless labor for this to function.
But, this would hardly be the first time a person tried to press culture to do a thing that doesn't actually work. :/