Okay, let's try to make this simple...
To work you need a SSN, the same as in other countries you need to show your ID number, etc. BUT since you are not a citizen of this country you need that and for that you need to show that you are eligible for doing so. There are many documents as you see on list A,B and C. List A and B are more for identity (remember you passport does not work since you are not a US citizen, must people do not have a valid US driver's license) and List C is for employment eligibility (SSN or EAD).
Social Security needs your EAD to validate that you need a SSN because you are ELIGIBLE to work. That's the only thing. You are not getting a SSN because you need one for other issues, you are asking SSA for a SSN because you want to work and for that you must show them that you have authorization to work in the US...that is why is called Employment Authorization Document. Check this, it's a document, it does not say that right away you can work with it as a an "only working card". It says that you are ELIGIBLE to work in the US.
Many people won't need to get a driver's license or do not want to (not everybody know how to drive you know) so as for identity issues then the EAD is perfect for that matter as well.
So in general yes you need your EAD and is an employment authorization document that you must use unless you have already a SSN (valid to work) and a valid US driver's license. Now, check this out...if you have a SSN under other provisions, you would like to update your records.