In terms of natural disasters, its actually up to each individual to have enough emergency supplies on hand to last at least a week. There is no government on earth who can supply a large city full of people with emergency supplies in less than a week, so each person has to take their own steps to care for themselves for as long as possible.
I personally would not leave my own care in an underfunded abstract entity over which I have no control.
For example, if I lived in a flood zone, you would bet I would have a canoe, and an inflatable boat, etc.
Or in a hurricane zone, a safe storm cellar, etc.
I think a good addition to this list would also be a full Emergency Kit in your car, in case of trouble on the road.
Other things of course would be a couple of small battery radios, lots of batteries, flashlights, candles, gas generator, etc.
It is occuring in the back of my mind, due to the media hype these days, that a serious flu pandemic, which could start next week, or not in our entire lives, is also something to think about. To be honest there is not much one can do about that, unless you have a large box of high quality masks, gloves, glasses, and are able to move to a remote area quickly. I think in that case, the best you can do is to cut down on the odds a bit.
Obviously the key aspect of this, is to go from the list, to actually having the preparations DONE, and then perhaps updated a couple of times a year.