The APIs are marginally profitable. You can calculate the lifecycle costs of the open models on clusters in batched inference and figure out its less than than what they charge.
The training and researches are very expensive. The fixed price subscriptions are 100% a sweetheart deal.
OpenAI for its part is tracking to $12-$15 billion in annual sales. If they slapped a basic ad model for referring onto what they're already doing, it's an easily profitable enterprise doing $30+ billion in sales next year. Frankly they should have already built and deployed that, it would make their free versions instantly profitable and they could boost usage limits and choke off the competition. It's the very straight-forward path to financially ruining their various weaker competition. Anthropic is Lyft in this scenario (and I say that as a big fan of Claude).
Because Anthropic is presumably massively subsidizing the usage.