Certainly depends on what you count as "disaster", but depending on where you are flooding, multi-day power outages, multi-day interruptions of fresh-water supply, ... all can happen. I yearly get flyers on what to do if the closest nuclear plant goes wrong. Italy just had a major earthquake.
Even if we assume the support infrastructure works properly (=just one localized disaster), having your own supplies could mean a lot of difference in how you get through the crisis:
Can you take care of an injury if medical help is going to take many hours to get to you (or if you are unable to even notify anybody who could help)?
Do you have to get to somewhere in the hope of getting some minimal supplies (water, some emergency food) or can you stay somewhere relatively comfortable and safe? Large cities might turn dangerous relatively quickly.
Can you have warm food despite the power being out the last few days?
Are you going to have a full set of warm clothes/medication/documents for everybody if you have to leave in a hurry?