I don't have a free phone from the government, but it is a prepaid phone, often called a "burner phone.", I've used it a long time, and I have an account set up with the provider, but not a name association in public records. I do it that way because this phone costs me about half what I would pay for a phone in my name through a contract with Verizon.
In my way of thinking, that should not matter anyway, because I am not required by law to have ANY phone, plus I had already submitted a picture of my drivers license and a selfie through the very phone they were questioning, as well as responding to texts in the sign up process. So they knew by that it was my phone.
So here we get to the crux of it.
No, you don't need to have a phone. You have one, and it did you no good at all. What you don't have is a long-term phone number association, which is part of the puzzle of confirming your identity.
This is hardly unique to the SSA. I've encountered this very same thing with utility companies and such when making service changes.
Not all prepaid phones are considered burners. Those involve the use of disposable phone numbers.
Nothing says you couldn't have gotten the new phone and had your old phone number transferred over to it. The regulation requiring this was not just for consumer convenience, but also to facilitate identification.
I've had prepaid phones for many years, but I always transfer the number when I change carriers. I'd had a plan with some limits that cost me $100/year,. When I came to need data as well as talk and text, I went another way for $240/year. After a year of that I realized I didn't need that much data, and so for the next 12 months I've paid $180 for the year for just 5GB/month instead of 15GB.
All the while bringing the same cell phone number forward with me.
I also got pushed out of "copper lines" for my home phone, and was transitioned to a cellular phone "box" that my home phones plug into. Again, I was able to transfer my long-time home phone number to the new "line."
The upshot is that your practices have eliminated one of the more common points of identity validation. One that works for more than 99% of people. You have to expect some inconvenience.