Yes. After making the FCC complaint, I was contacted within two days by a Spectrum rep who assured me that new Spectrum plans still support cable cards in Cincinnati. With her email in hand, I went to the closest Spectrum store. The agent at the desk that I talked with first said that I wouldn't be able to use a cable card on a new plan. I showed him the email and he went to the back to talk with his supervisor. The supervisor came out and said that yes indeed I could move from my legacy TWC plan to a new Spectrum plan and keep my cable card.
The supervisor showed the agent how to enter the order into the system, and was literally getting ready to walk out the door when the agent said that he was getting an error code when he tried to submit the order. The supervisor came back and tried a couple things on the agent's terminal which didn't work, and then saved the order and moved to another terminal where he logged into his own account at which point he could submit the order. So it appears that there are user security settings that prevent regular Spectrum agents from submitting orders that include cable cards.
Edit: I also moved from 20Mbps to 200Mbps download, and got a new modem. After I self-activated the new modem, I was still getting only 20Mbps download so I called Spectrum. The agent said things looked like they were setup correctly, but sent a signal to the modem. Shortly after that I started getting 200Mbps download. So I don't know if what the agent did made the difference, or if it was just a matter of time (about 30 minutes after activating the new modem) that did it.