Space is really big, though. If you're in the asteroid belt, it doesn't really help if the Space Coast Guard is also in the asteroid belt because they might still be multiple AU's away.
Consider that the delta V from Mars to Earth is 15 km/s, and we use gravity boost to make that delta V.
Assume we invented engines that could manage that delta V in half an hour. If my math is right, if you wanted enough 'fire departments' to reach every asteroid within one hour, you'd need about 67,000 of them, and you'd have to figure out how to pull that delta V without running into any other asteroids (probably out of the ecliptic?)
The good news is your paramedics would only pull about .8Gs which, while perfectly comfortable for earthers would still be tolerable for belters. My point is that if you try to decrease the number of stations to, say, 100, you're going to turn any human occupants into a fine mist in the process.
Did you look at any of my math at all? 1/67000 is a pretty tiny fraction alright.
Ok, here’s some more math. The average distance between asteroids in the belt is apparently over twice as far as the moon. The moon is still very far away for us. There will be no rescues for a very long time and remember that humans die at 10Gs, so when I say very long time I’m not just talking about faster rockets. I’m talking about basic research in human physiology.
More NASA videos and The Expanse for you, less Star Trek.
This is not exactly true. Arguably, it's half false. In any case, this is more nuanced than what you state. There may be some humans who die at 10G. Prolonged exposure to high G might increase the death toll. People could be maintained at 10G unconscious, if they are in the right position, on the right kind of couch.
I was about to argue with you and then realized I misspoke. Humans begin to die at 10Gs, as in survivability starts to drop at that point and if the point is to get the humans there and back again, that’s not very good.
Regardless, if people can survive 20Gs that still only doubles your range. And since we are discussing a category error of orders of magnitude, my point still stands.
That's true, but in all likelihood companies will cluster their operations together for this very reason. It's worth sharing some kind of common infrastructure in a specific location to protect workers/assets.
Besides, a few days/weeks is a better response time than a few months.
Sure, but then you need at least two central points for the Space Coast Guard to operate from: Earth itself (inevitably) and whichever "cluster" in the asteroid belt you want.
And there's really a limited size of an asteroid belt cluster you could get. If you start from Ceres and go a few hundred (even dozen!) meters either towards or away from the sun to reach another nearby asteroid, that asteroid has a completely different orbital velocity and will not line up with Ceres again until either it or Ceres completes another full revolution. That means you have to be in essentially the same orbit and travel either clockwise or counterclockwise, which is most efficiently done by making a more eccentric orbit (so your aphelion is the same as Ceres' but your perihelion is lower, or alternatively your perihelion is the same as Ceres' but your aphelion is higher) and then readjusting once you reach the target. But even this maneuver requires going the long way around the sun. Similarly if you want to go up or down relative to your orbital plane (i.e. change inclination).
If you want a cluster of bodies in space that stays relatively the same distance from each other, you have to have a smallish orbital system. The Earth/Moon system is one of these, and it's going to be a lot more feasible to capture near-Earth asteroids and stash them in Earth orbit than to try and find a meaningful cluster of belt asteroids that actually stay close together over the long term.
Being close in time is still a factor for manned missions when starting out. If something goes very wrong, rescue is only a few days away.