No.
You can issue the form 1099-NEC as an individual using your SSN.
The reason you may want to apply for an EIN is to avoid disclosing your SSN to third parties.
Why? What are you trying to accomplish? Who did the caregiver take care of? If the caregiver provided care for a dependent you claim on your tax return, the caregiver may have self-employment income or may be considered to be a household employee of yours--depending on where the care took place. But if you are not a business you do not need to issue a 1099NEC to someone who was paid for services. If the caregiver was a household employee you issue a W-2 to them. If they worked as an independent contractor then all you needed to do is enter their information for the child and dependent care credit and use their SSN or Tax ID. Reporting their self-employment income on their own tax return is their responsibility.