Our 5.5y pug had died after his anesthesia started taking effect. He was there for Brachycephalic Airway Syndrome, and seemed healthy otherwise. He had removed few bad teeth 4 months ago with all the blood test being perfectly normal. On clinic, they did "full workout" testing and x-rayed his chest. All fine. The theory is, he was living with heavy oxygen debt and anesthesia slowing him down was simply too much. But looking at the invoice and quick call back there, the clinic didn't do any blood tests. They claim it's not a standard procedure and while I understand my knowledge is limited, it's killing me that could perhaps show something and prevent him undergoing the anesthesia.
How important a blood work is? Could it warn about low oxygenation?
EDIT: Clinic has explained a blood testing was deemed not needed according to their extensive practice and what they learned about him from other tests. Their original "he seemed healthy" was indeed backed up strongly when described in detail. They did experience a few similar cases but there's still no clue what are the indicators to look for, except all were brachycephalic dog breeds. Blood test wouldn't help with oxygen level. (see the accepted answer)