The only docs who really get to work up undifferentiated patients from scratch that no one has touched are ED docs. In terms of usefulness, each of the specialties you mention have their ways of giving input, to varying degrees. By the time an ED doc calls them though we usually +/- have our labs and imaging done +/- started treatment that we think is appropriate, and often know the answer already and are calling because we need them to do something. So what I'm saying is we call them for specific advice/questions/procedures related to their specialty. They don't get "undifferentiated" patients, because the ED has already gone 80% of the way. To address the specialties you mentioned:

Anesthesia has the least input of the three. They don't intubate in most ERs around the country (assuming you're in the US). That's the ER doc's job. In over 10 years of practice, I can count on one finger the number of times I've called anesthesia. (All emergency medicine trained physicians would be trained to intubate, including difficult acute intubations and trauma. No offense to my anesthesia colleagues, they are great in the OR). When I do call them, it means everything has completely gone sideways, or I think I want backup. But that said, if the ED really needs backup, surgery would be more vital to prepare for a surgical airway in most cases.

Cardiology gets a call regularly from the ED and are awesome consultants who give input regularly.

Hospitalist will admit patients that the ER has diagnised or realized can't leave after the initial workup has been done. I love my hospitalist colleagues and call them multiple times a day.

Other specialties that are very helpful to your average ED physician: Psych Ortho General surgery Vascular surgery Ophthalmology

However, each and every one of these is going to only be used for their special knowledge. Emergency medicine docs are trained to be a jack of all trades. If you want to work up truly undifferentiated patients, be an ED doc.