When you fill out the Schedule C, you are able to use your EIN (if you have one) as well as your business name. The 1099-MISC does not have to be issued in the business name if you are a sole-proprietorship. You can input the information that is on the 1099-MISC as it appears on the form and it will transfer over to your business income reporting on Schedule C.
The business income reported on Schedule C ultimately is linked to your SSN. So a 1099-MISC issued in your name instead of the business ends up in the same place in the end.