Define profit? If by profit you mean covering compute costs then yes you're correct. If by profit you mean covering spent RnD budget and future RnD budget allocations then no, by no means are any of these AI companies running at a profit.
Also these third-party providers are running magnitude smaller models than what Anthropic and OpenAI's power users expect from them. Yes the average chat message isn't being routed to the 1+ Trillion param model but people running CC or Codex on Ultra probably are. Those are objectively more expensive.
I am citing out of my head here, but think deepseek’s inference engine with their models create around 30 dollars per hours while it costs around 4 dollars, there might be other costs but if you look at the demand, it will become very profitable. I don’t understand why people are pessimistic about the profitability (I don’t want them to be this successful either). BUT why would they care about profitability right now. They believe in AGI, so do the big money people. They get their investments. This train is not stopping and I believe their predictions are correct (IMO we should not debate about timescales apart from spreading urgency). And for the lowering of prices, it is possible because they just can and hooking people now into their systems will be 10x more profitable in the future. In addition to all the juicy training data they get with real world clients. They are scared the others win, they are not scared that they run out of money.