Yes, many people believe that, but it doesn't seem to be an evidence-based belief. I've written about this in some detail[0][1] before. But since just linking to one's own writing is a bit gauche and doesn't make for a good discussion, I'll summarize :)
1. There is no point in providing paid APIs at negative margins, since there's no platform power in having a larger paid API share (paid access can't be used for training data, no lock-in effects, no network effects, no customer loyalty, no pricing power on the supply side since Nvidia doesn't give preferential treatment to large customers). Even selling access at break-even makes no sense, since that is just compute you're not using for training, or not selling to other companies desperate for compute.
2. There are 3rd-party providers selling only the compute, not models, who have even less reason to sell at a loss. Their prices are comparable to 1st-party providers.
3. Deepseek published their inference cost structure for R1. According to that data their paid API traffic is very lucrative (their GPU rental costs for inference are under 20% of their standard pricing, i.e. >80% operating margins; and the rental costs would cover power, cooling, depreciation of the capital investment).
Insofar as frontier labs are unprofitable, I think it's primarily due to them giving out vast amounts of free access.
[0] https://www.snellman.net/blog/archive/2025-06-02-llms-are-ch...
[1] https://news.ycombinator.com/item?id=44165521