Garmin Response during a natural disaster?

Yes

One other note here. You may (or may not) want to look at the "insurance" mentioned in the article linked from Leslie's post. In the US, there are usually few if any "people" costs associated with SAR. SAR is mostly staffed by volunteers who donate their time. However, there may well be hard costs - for example, if a helicopter evacuation is involved, or if an ambulance transports you from trail head to hospital. 

it can also depend on what agency provides the response. US government agencies (for example, Coast Guard or NPS) do not typically charge individual citizens (your tax dollars at work). The situation may well be different if state resources are involved. And there is always the issue of whether or not you were "reckless" or "negligent". For example, if you ignore posted "trail closed" notices, you might get billed for SAR when there otherwise would not be any cost.

Outside the US, policies vary considerably. But costs are more likely.

Bottom line: It's complicated. Your standard inReach plan ONLY covers communication costs. It does not cover SAR or subsequent costs.

During a recent flood caused by a tropical storm, our regional 9-1-1 emergency phone system went out of service due to a cut in the fiber-optic cable. My question is, in a widespread natural disaster such as this, will Garmin Response relay inReach SOS Messages to the appropriate local agencies? (This question only concerns emergency communications, not insurance.)

I think they will try, but because of the mentioned scenario, they may not reach them, if the communication system of the local agencies are effected through the natural desaster and they will also be heavily overloaded locally if a larger region is affected. It will take time to coordinate this, if a bigger region is affected and I think the local agencies will create a list of priorities that they will try to work through. But that are only my thoughts