While defects ordinarily reduce yields, Intel put plenty of redundant transistors into the silicon. This is ordinarily not possible to estimate, but Tom Petersen reported in his interview with hardware unboxed that they did not count those when reporting the transistor count. Given that the density based on reported transistors is about 40% less than the density others get from the same process and the silicon in GPUs is already fairly redundant, they likely have a backup component for just about everything on the die. The consequence is that they should be able to use at least 99% of those dies even after tossing unusable dies, such that the $57 per die figure is likely correct.
As for the rest of the card, there is not much in it that would not be part of the price of an $80 Asrock motherboard. The main thing would be the bundled game, which they likely can get in bulk at around $5 per copy. This seems reasonable given how much Epic games pays for their giveaways:
https://x.com/simoncarless/status/1389297530341519362
That brings the total cost to $190. If we assume Asrock and the retailer both have a 10% margin on the $80 motherboard used as a substitute for the costs of the rest of the things, then it is $174. Then we need to add margins for board partners and the retailers. If we assume they both get 10% of the $250, then that leaves a $26 profit for Intel, provided that they have economics of scale such that the $80 motherboard approximation for the rest of the cost of the graphics card is accurate.
That is about a 10% margin for Intel. That is not a huge margin, but provided enough sales volume (to match the sales volume Asrock gets on their $80 motherboards), Intel should turn a profit on these versus not selling these at all. Interestingly, their board partners are not able/willing to hit the $250 MSRP and the closest they come to it is $260 so Intel is likely not sharing very much with them.
It should be noted that Tom Petersen claimed during his hardware unboxed interview that they were not making money on these. However, that predated the B580 being a hit and likely relied on expected low production volumes due to low sales projections. Since the B580 is a hit and napkin math says it is profitable as long as they build enough of them, I imagine that they are ramping production to meet demand and reach profitability.