First, who says it is not used?
The US National Guard (which is a branch of the USA Armed Forces) is regularly deployed in emergency situations, like for example the Katrina hurricane. And, AFAIK, other branches of the Army can be deployed if the situation is needed.
An important difference between National Guard and the Federal Army, Air and Navy forces is that the Posse Comitatus Act severely limits the roles for the Federal forces1, banning them from being used to enforce laws2 in most of the situations.
To answer why don't you see members of the armed forces immediately at each emergency scenario, a couple of thoughts:
National Guards and other units, as part of the Army, are outside the control of civilian agencies and are activated following their own chains of command. That means that the local PD or FD chief or even the major cannot just call the NG barracks nearby to provide manpower, they have to ask the Governor or POTUS to activate them. The time for this procedure means that only a few emergencies are big enough for that.
The "fog of war", in an emergency often people need some time to get the info about how difficult the situation is, which resources are available, etc. delays the call for help.
The people in charge may fear that calling for help makes them appear in public as unready or uncapable for handling the event. This may cause such people to delay asking for support, in the hope that the emergency improves without external help.
considering that almost all military personel have VASTLY better training compared to the average policeman.
Usually, military personel have very good training... for waging war. They sheldom have training for managing crowds or firefighting, they do not know what are the operatives, duties and rights of law enforcement officers, they don't have materiel for crowd control or for reducing someone without lethal force.
Their best asset usually is that the are a quick source for manpower and/or equipment (trucks, helicopters, etc.), but using them for keeping order is a significant risk of armed incidents. Sometimes NGs are deployed without weapons to prevent this kind of incidents, sometimes as part of mixed patrols with cops and soldiers.
It would also make sense for foreign diplomacy to have the US military have help with disasters across the globe.
AFAIK it happens, but the procedures above mentioned become even more complicated.
The foreign government must have a need for it; if there is an emergency that government forces would probably already be fully activated and providing most of the manpower needed.
The foreign government must request3 (or at least accept) USA support and the POTUS must agree and activate the units, lengthening the process.
The security of the emergency personnel must be reasonably ensured, which makes if difficult for deployments in countries with the worst humanitarian crisis (civil wars, etc.).
Coordination must be stablished with local authorities; this is even more complex than in the USA because most of the USA soldiers will ignore the country language, laws and cultural differences.
All of the above means that usually the relief help accepted is limited; usually specialist units (medics, rescue personnel, air transportation), emergency supplies and perhaps stablishing logistic bases.
1 It is worth mentioning that the Enforcement Acts still allow for the POTUS to use Federal troops into a State that refuses or cannot control attacks on constitutional rights.
2 Appart from that, nothing appears to limit the use of Federal troops (if properly activated) to perform duties not related to law enforcement, like digging an anti-fire trench or distributing food.
3 And there may be reasons of national pride/prejudice/propaganda against requesting or accepting such help, even if needed.