Meanwhile getting rid of garbage men would require every American to deliver their own bags of trash to the dump. This is still common practice in many parts of the US.
Garbage men also manage the dump. Instead of an organized system that manages waste and prevents problems with run off, you'd just have people who were willing to throw garbage somewhere on their property for cash.
How many people would bother taking their trash to the dump instead of somewhere closer? (illegal dumping is already a problem now with pickup).
Yes, professional teachers do a very important job, but there's a reason our society did just fine without mandatory school laws for so long. Yes, there would be horrible consequences for not having public education, but it's a hierarchy of needs. Health trumps education. Sick kids don't learn well even with great teachers.
(and if you want an even better example, use sewage treatment workers....)