The training costs are well below the profits for each model.

That's the profit margin on inference, not overall. Each model does end up being profitable over its lifetime, but the money they're making is being immediately churned into buying more data centers & the training for the next giant model up, so they're not profitable overall at the moment. It's a bit like how Amazon kept churning their profits into more growth instead of taking the profit early.

That said, Anthropic has supposedly crossed over into profitability and made $1 Billion in profit so far this year, in the lead up to their IPO. Being profitable sounds good for launching on the stock market! But as a customer, that's noticeable in the downtime due to lack of compute, and only getting 50% access to Fable.

OpenAI might not be profitable, but they've got so much compute access that they've been able to give their customers full access to Sol, and as a result they've almost doubled their Codex subscriber base in the last two weeks (6 million on July 12, 10 million on July 21 - that would be an extra $1-$10 Billion in Annual Recurring Revenue that they've gained in just these 2 weeks). Doing the unprofitable thing in the short term can result in outsized rewards in the long term.