Unfortunately, AM stations are the most resilient in the case of emergencies. Even with a fallen tower, they can be put back on the air with a length of wire between two phone poles or even trees. When run non-directionally, they often cover far better than FMs, and in the event that all of a market area's stations are off, AMs from outside the market can penetrate, at least at night. FM is less resistant, and digital TV is useless when power is out. Web based news sources have an even weaker sustainability in emergency situations.
Another problem is the lack of economic viability of many AM stations. Fewer and fewer have their own news staff, so even in an emergency they are at a loss for a source for news unless they have an arrangement with a TV or newspaper outlet.
AM is useless during a hurricane. The lightning cancels the carrier on every other word. I was trying to listen to KTRH during Hurricane Ike on a battery-powered radio and ended up switching to ABC 13 audio simulcasting on KMJQ.
During Harvey, I noticed that KUHF was simulcasting KHOU and KTRK audio uninterrupted on their HD subchannels. KTRH was hard to listen with the lightning strikes on every other word again. The other stations did not have dedicated news staffs and Cox's Houston coverage did not last for more than a few hours. Even KTRH pressed a sports talk host from a sister station into anchoring duties.
Luckily Harvey was a rain event, unlike Ike, so power was still on and everyone was mostly watching TV (for 3 days straight). But even with power out and a battery-operated radio, FM is still accessible as long as the TV/FM towers are still standing. In the aftermath, AM becomes useful again with the lack of lightning but surprisingly the Missouri City antenna farm handled Ike's winds very well.