Cook had played every possible game for NSW since debuting in 2018 but coach Brad Fittler opted not to select two specialist hookers for Game I in Adelaide, naming only Wests Tigers rake Api Koroisau in his side.
Cook's surprise snub comes despite the Rabbitohs entering the Origin period with the equal-most wins of any NRL side, and Cook playing every minute of their season so far.
The Rabbitohs' biggest loss of the year came on the eve of Origin selection, the side falling 36-16 to Parramatta on Friday, which Demetriou said likely scuppered Cook's hopes of retaining his spot.
"He's disappointed but that's on us as well," he said.
"We didn't play well last week. We needed to play better than we did and blokes like Junior (Tatola) and Cookie and Stretch (Campbell Graham) would've put themselves even more in the frame.
"As a coach and as a club, we've got to wear that as well."
Graham had been considered a strong chance to make his NSW debut in the centres but was pipped at the post by Tom Trbojevic and found himself named 18th man.
He subsequently pulled out of Blues camp with a long-standing sternum injury, which flared up on Monday and meant he would have been unable to train early in the week.
The Rabbitohs were hopeful he could line up on Saturday night despite pulling out of Origin camp but ruled him out on Friday.
"It had nothing to do with Origin. He just woke up on Monday and Sunday a little bit sorer than he has in previous weeks," Demetriou said.
"He hasn't had any training this week and that was probably the biggest reason why we're leaving him out.
"He has been able to train in previous weeks but he was struggling out there to catch a ball.
"If we needed to we could've pushed him and needled him but he's been pushing himself pretty hard for the last 11 weeks. It's a good time for him to get that week off."
Demetriou predicted 23-year-old Graham had a long representative career in front of him despite missing the series opener.
"He'll play Origin, there's no doubt about that," he said.
"I think he's done enough to get picked but they've got some incumbents that have done the job for them before.
"When he gets it I'm pretty sure he'll hang onto it for a while."
