Nearly all are run by companies making money by recording as much of your data as possible.

A commercial cloud app could be entirely private but then the company would not be able to take your data.

"The cloud" is just somebody else's computer.

Why would they be? Clearly no one cares about privacy, and they can sell your information to make money.

There is no real technical reason. They could be. It's just a matter of encryption while in transit and no logs. Proton does it.

The real reason most aren't, is that there's not much incentive to make them private. The data has potential value, so why not hold on to it.

Because the companies that provide these models/apps aren't doing a service to the world. Their only motive is to make as much money as possible. I think that this is their idea to achieve that:

1. Create an AI model that's excellent and free.
2. Users use it and since they like it, they continue using it.
3. Use the chats to train the next generation of models and make it more powerful.
4. Ensure that a lot of people use AI so frequently they get addicted to it because they feel its so good and useful.
5. Once enough market share is captured, place everything behind a paywall.
6. Everyone who is addicted to AI will pay because they can't live without it.
7. Profit

At least, this is probably Sam Altman's dream.

What do you mean by private? If they're non-self-hosted, then they're not private by definition.