Didn't his program go offline when the computer was down? Hence the introduction of the mobile emitter allowing him greater access to the ship and being independent of the ship.
Two unrelated things but yes: he is a self contained program stored on the computer. If the computer goes down, the Doctor goes down. That doesn't mean that he is the embodiment of the entire computer though. Think of it this way: you download a new app on your phone. It wants permission to access A, B, and C on your phone. The Doctor does not have access to A, B, or C. He has to get it like everyone else.
Second, this is not the reason for the mobile emitter. The ME was introduced because they had simply reached their limit with what they could do with the character restricted to sickbay and the holodeck.
I never said the Doctor was the embodiment of the ship's computer, but rather, to use your app analogy look at it this way. My Facebook app has access to my camera, gallery, and contacts it needs them to post pictures I want it to and to send invites to people not on Facebook or look up friends Facebook accounts. So the Doctor and all EMH programs must therefore have acces to at least the ship's manifest, medical supplies list, sickbay computers, biobeds and other data.
Like in the example I gave:
EMH: Please state the nature of the medical emergency.
CMO: Victims of a comet impact on planet Beta 5 are being brought on board. Man station "C."
EMH: Understood.
By being able to access the ships computer the EMH can, determine what type of injuries it would most likely encounter, the species it's going to be dealing with, what to do, where to be:
1. comet impact - smoke, dust inhalation, blunt force trauma, crushing from being trampled by running people or falling buildings, burns, and more
2. planet Beta 5 - what different species inhabit Beta 5, what medical treatments that can and cannot be used on them
3. man station C - where he needs to be
1 & 2 he'll get from the computer without having to ask unnecessary questions. Being able to get the biobed's readings without a medical tricorder frees his hands to work while the biobed gives him the patients diagnostics and where and what injuries he must treat in order of importance. Being able to remotely use the surgical arc that come over the patient allows him to multitask. Something the Doctor on Voyager should have been doing.
The "E" in EMH IMHO is not just if the ship has an emergency. For instance, a patient in an isolation ward can be treated by an EMH since it is immune to disease. The 1701-E had one as well, which makes me think it was also designed to be another pair of hands...
I value personal opinions, but in this case it goes against established fact and intention. The "E" in EMH does mean "Emergency" because this is what it was designed for and intended to be used for - Short-Term relief during emergency situations. It was installed on all starships for emergency purposes. If Crusher "needed another pair of hands," it's probably because she and her staff were overloaded, hence they are in the middle of a crisis.
Yes the EMH would be used n crisis, but not necessarily one where the ship has been damaged. Providing planetary relief would go better with the EMH working.
I'm not saying that the EMH idea couldn't be used for other applications, like in FC Crusher improvised and used the EMH as a distraction. But you're not going to find an isolation ward on a typical starship.
Sure at a starbase, colony, etc., they may have another version of the program suited to perform isolation/ultra-sterile services and operate for longer periods. But those other versions probably wouldn't be called EMH.
On a Starship it's called an EMH because that's what it's indended to be.
But we do find isolation wards on starships. The Enterprise D was able to put up containment fields around biobeds and in "Brothers" the boy with the parasitic infection was in an actual isolation chamber. Voyager had an isolation ward as well, Torres was in one when that scorpian thing attached itself to her. If a Galaxy class and Intrepid class have isolation wards or fields then it is entirely possible that all ships have them. You never know what you're going to encounter.