"notifies the appropriate rescue agencies"
That is the tricky part. Based on the GPS position sent in the SOS message, somebody has to look at a map and figure out who to contact. Inside a national park, they have to contact the national park dispatcher. Then somebody else has to look at a map and figure out whether to call out SAR or somebody else. That is where the value of a two-way satellite communicator really shines. The dispatcher can contact the victim directly to try to better assess the emergency:
"What do you mean you have a broken leg?" and "Can you wait twelve hours for helicopter evacuation?"
Outside the national parks, it is often a county sheriff that has responsibility, so the sheriff's dispatcher gets the call. Again, somebody has to look at a map and figure out what to do. Then in other jurisdictions, it might be some other authorities. Only those legal authorities then contact SAR, if that is appropriate. It would be indeed rare that SAR goes on a rescue on their own.
I recall the emergency that happened around Mount Whitney back around 1991. Mount Whitney is right on the edge of a National Park and a National Forest. Backpackers entered the summit hut to avoid a storm. Lightning hit the hut, and one person was killed outright. Others were severely burned. They had no help at all, so one strong backpacker ran down the trail to the 12,000 foot level where a bunch of backpackers were camped, and he was screaming for help. There were some Boy Scouts there, and one guy had an amateur radio. Once he understood the emergency, he flipped the radio over to the aircraft distress frequency and started transmitting a Mayday. An airliner was passing overhead right then on its way to LAX. The pilot got the message and relayed that to LAX. There at LAX, they looked at a map, then called the Inyo County Sheriff's office even though technically it might have gone to Sequoia National Park. The sheriff's office could not do anything immediately, so they called the Air National Guard. They flew a small helicopter up to the summit and plucked off only the two severely injured people. It could not lift anymore than that. The two injured were rushed directly to the hospital, and then a large helicopter was flown up to the summit early the next day to get the rest of the backpackers and the one deceased. The moral is that all of this rescue coordination stuff is not as simple as it might appear on paper. So, think twice before you hit that SOS button.