Am I required to provide my SSN to a third-party payor in this situation?

Let’s just say that TPP wants your information as the contact or representative independent of any consulting you do. Then wouldn’t they want your contact information and the company’s EIN? You said that in your OP but that should be in lieu of your SSN not in addition to it.

Yes, that’s my opinion. They can have My Company’s EIN, address, contact information, annual corporate report, etc., but they don’t need mine. If they have mine, how can I tell if they are going to list it as the company address somewhere? What if they decide to use my SSN instead of My Company’s EIN when they file their tax info? What if a 1099 shows up in my mailbox next January for the payments they made to My Company in 2024?

Do they somehow mistakenly think you are operating as a sole-proprietor or DBA?

I don’t know. They have requested my SSN, DOB, and mailing address at minimum. They then state that I can “add” an EIN later if I choose to. I suspect, therefore, that they are assuming I’m a sole proprietorship or a DBA. To be fair, most of the other instructors are probably either doing this work personally as an independent contractor or as a sole proprietorship, but I’m definitely not.