This made me wonder whether it would be possible to build a Wi-Fi-only, roaming-only carrier for computers.

Your carrier is already capable of redirecting your SMS messages to other carriers, that's what they do when you're abroad and roaming with a foreign operator. You could make a fake carrier that speaks the right protocols on the roaming side, but communicates with the customer over the internet (using an API or a proprietary app) instead of LTE or GSM.

This would essentially work like an SS7 redirection attack, but with the full knowledge and consent of the "victim." You could alleviate the security impact here by requiring SIM card authentication, just like a normal carrier does, which could be performed through the internet and an USB reader just fine.

Carriers would probably hate this and might not be willing to sign roaming agreements with such a company. I wonder whether a gray-hat route would be possible here, especially if the company was outside US jurisdiction.