If the model companies are serious about demonstrating the models' coding chops, slopping out a gmail competitor would be a pretty compelling proof of concept.
In that case, there's some ancillary value in being able to claim "look, we needed a gmail and ChatGPT made one for us - what do YOU need that ChatGPT can make for YOU?"
Even at our small scale I wouldn’t want to be locked out of something.
Then again there’s also the sign in with google type stuff that keeps us further locked in.
The challenge in migrating email isn't that you have to move the existing email messages; any standard email client will download them all for you. The challenge is that there are thousands of external people and systems pointing to your email address.
Your LLMs memory is roughly analogous to the existing email messages. It's not stored in the contacts of hundreds of friends and acquaintances, or used to log in to each of a thousand different services. It's all contained in a single system, just like your email messages are.
However, I take your point - OpenAI has an interest in some other party paying them a fuckton of money for those tokens and then publicly crediting OpenAI and asserting the tokens would have been worth it at ten fucktons of money. And also, of course, in having that other party take on the risk that infinity fucktons of money worth of OpenAI tokens is not enough to make a gmail.
So they would really need to believe in the strategic necessity (and feasibility) of making their own gmail to go ahead with it.