You implied it, so I offered two real-world use cases from a SPOT and a Delorme (now Garmin). Both times "the big boys showed up". In fact, your statement is wrong when you state that it depends who shows up when using a Delorme. As I run an avalanche center, we researched this to understand and consider what made sense for us and our clients in the backcountry during certification activities.
Pat Caulfield, the Vice President of the Colorado Search and Rescue Association and Commander and Team Commander of Fremont, Colorado Search and Rescue states that when a signal is initiated from a satellite messanger (Spot or Delorme), they "send SOS calls to the GEOS International Emergency Response Coordination Center in Montgomery, Texas, regardless of where they are activated". Furthermore, Emily Thomson, the Operations Research and Development Manager at the GEOS IRRC states, "When someone activates the SOS on their satellite messenger it will come into our monitoring system with their profile information and coordinates. Since it’s a two-way device we’re able to communicate with them about their emergency, and as soon as we receive the alert in our system with the coordinates, we will immediately dispatch emergency services based on their location.”
She goes on to state that, “For the United States,
we will coordinate with the county sheriff department, or if they are in a national park, we will coordinate with the national park dispatch,” Thomson explains. “If it’s international, it will be the rescue coordination center for the country that the device activated was in, so we will coordinate the rescue with them.” The bold font is my highlight.
So based on my research on the topic, my dual real-world experiences, along with this info, I'm fairly certain we can state that "the big boys" will always respond. Here's my full source.
Hit ‘SOS’ on Your Sat Messenger or PLB? Here’s What Happens Next