Why are emergency service vehicles necessary?
Something might have been damaged. What if something was and there is a fuel leak somewhere? Hydraulic leaks are also dangerous, and more likely as hydraulic lines run to the brakes and nose wheel steering. Better safe than sorry, so emergency services go check.
Sometimes, they also disembark passengers via the ladder if it looks like towing the aircraft to terminal will take too long and the aircraft is on unpaved surface where the mobile stairs have trouble going.
They can also help with moving the plane, especially on smaller airports that only have one runway and so have to close until the plane is moved away anyway.
I would think the plane is more than capable of getting back onto the Tarmac, so why doesn't it?
No, it isn't. Aircraft not designed for unpaved surface will usually sink in somewhat, especially if the ground is wet, and require a lot of power to pull out. Sometimes, the tow truck is enough to get it out, sometimes they have to put planks under the wheels and sometimes they have to get out hydraulic jacks or inflatable support bags.
Even if the plane did not dig in and is easy to move, providing the power with the engines is not a smart thing to do, because the engines will throw up dirt and stones that could damage them, other parts of the aircraft, or something else around. Tow truck is much better for that job.