As EMTs we are trained to provide the best possible care for all of our patients unless very obvious signs of death are present - brain death is not something we really consider because that's something the hospital determines. Obvious signs of death include dependent lividity (blood settling to the lowest point in the body) and decapitation, among other things. EMTs are not qualified to make such judgements in the field (outside of a mass casualty incident and triaging) and opt to be "better safe than sorry" and let someone with an MD after their name and malpractice insurance worry about making the final call. I expect the signs and symptoms of someone with brain death would be indistinguishable from any other apneic unconscious patient without a pupillary response, but some of those can be saved, so we transport them all.

At the hospital, protocols for determining brain death vary depending on the institution, but we have more advanced tools than just EEG now. I see a lot of radiological investigations that measure cerebral metabolism with scintigraphy - here's an example of what that looks like with a poor outcome (I am not a radiologist by any means, and I explored this outside of being an EMT) [1]. Notice how the brainstem is dark but the rest of the brain isn't doing much. Other investigations such as some MRI sequences may prove useful to identify ischemic insults and damage that strongly correlate with poor outcomes. Based on the phrasing of your question, I would point out that these tests would only be performed on a patient where brain death were clinically suspected based on e.g. prolonged cerebral hypoxia from a drug overdose... not every dying patient.

I don't know enough about the medicolegal or surgical side of things to go into more detail there.

[1] - https://img.medscapestatic.com/pi/meds/ckb/59/26659tn.jpg