Selling inference for more than inference costs is not incompatible with bleeding cash at ungodly rates. They do in fact pay ungodly amounts of cash for other things, like training, marketing, etc. Heck, you can bleed cash while being profitable (in the accounting sense)

Also, API prices going up a lot every new version is more an OpenAI thing, and even there it's a recent trend: GPT 5.0 was a big price drop compared to 4.1, and 4.1 was cheaper than 4o, which itself got a price cut at some point and is cheaper than 4. Meanwhile Anthropic's API pricing stayed stable for many versions, then got slashed to a third with the 4.2 release and have stayed at that level since.

I am conveniently not counting training costs (since they add no marginal costs, selling more tokens doesn't impact them), and hardware and DC costs only amortized

Of course they do have to "make bank" in some way to offset the insane training costs. But whether they go for high prices or high volume, or offer some services as a loss leader to drive profits elsewhere is somewhat orthogonal to that