A death certificate is your work permit

An undead zombie must present a valid death certificate in order to be legally used as labor. That certificate lists things like cause of death and whether that body is licensed for labor use. Using unlicensed undead labor is a serious offense that will result in you losing your business, plus ensures that your corpse will eventually spend eternity toiling away in the worst, most grueling jobs available. Bodies that are not licensed for work will have the major ligaments in their arms, hands, and legs severed so that even if someone tried to reanimate them, the body wouldn't be usable at the mechanical level.

Deaths resulting from criminal activity or suspicious circumstances are automatically refused work licenses, so no killing people to boost your workforce. The cause of death is documented in detail, so any potential employer (is that even the right word any more?) knows exactly what the limitations of the body are. Deaths caused by injury are likely to leave the body damaged and unable to work completely, so a shady employer can't skimp on the workplace safety practices to kill off employees and then work them for free after they're dead. The type of zombie valuable for work purposes is the kind that died naturally and peacefully.

Your fears about the benefits of a high mortality rate are mostly unfounded. Unlike living workers, the undead don't retire, relocate, or need replacement. They're mindless, so the types of work they can do is limited. Together this means that you'll relatively quickly reach a point where all the work that could be done by the undead is being done by the undead. The demand saturates and excess deaths at that point aren't worth very much. Not to mention, a job that can be done by the mindless undead is a prime candidate for being automated out of existence as technology advances. Even though the undead are almost free, large portions of them will end up destroyed and replaced with machines that are much faster and more capable, precise, and consistent. Anyone still relying on undead labor will lose out to those that have mechanized. Undead labor will grow obsolete even faster than regular manual labor.

A death certificate is your work permit

An undead zombie must present a valid death certificate in order to be legally used as labor. That certificate lists things like cause of death and whether that body is licensed for labor use. Using unlicensed undead labor is a serious offense that will result in you losing your business, plus ensures that your corpse will eventually spend eternity toiling away in the worst, most grueling jobs available. Bodies that are not licensed for work will have the major ligaments in their arms, hands, and legs severed so that even if someone tried to reanimate them, the body wouldn't be usable at the mechanical level.

Deaths resulting from criminal activity or suspicious circumstances are automatically refused work licenses, so no killing people to boost your workforce. The cause of death is documented in detail, so any potential employer (is that even the right word any more?) knows exactly what the limitations of the body are. Deaths caused by injury are likely to leave the body damaged and unable to work completely, so a shady employer can't skimp on the workplace safety practices to kill off employees and then work them for free after they're dead. The type of zombie valuable for work purposes is the kind that died naturally and peacefully. Killing people to expand your undead workforce ultimately isn't practical because the zombie doesn't have the same capabilities as the living worker. They're cheaper, but don't generate anywhere near the same economic output. The only way they make economic sense is if you're repurposing something that would normally go to waste (the natural dead).

Your fears about the benefits of a high mortality rate are mostly unfounded. Unlike living workers, the undead don't retire, relocate, or need replacement. They're mindless, so the types of work they can do is limited. Together this means that you'll relatively quickly reach a point where all the work that could be done by the undead is being done by the undead. The demand saturates and excess deaths at that point aren't worth very much. Not to mention, a job that can be done by the mindless undead is a prime candidate for being automated out of existence as technology advances. Even though the undead are almost free, large portions of them will end up destroyed and replaced with machines that are much faster and more capable, precise, and consistent. Anyone still relying on undead labor will lose out to those that have mechanized. Undead labor will grow obsolete even faster than regular manual labor.