That's assuming that there's an extreme case present which is completely out-of-the-ordinary for most people, aside from those living in disaster zones. You're not wrong that there are far better investments for an emergency, especially a more extreme emergency, but it's not something that most people would be considering on most days.
At the end of the day it really is Time > Bitcoin (alt Gold, depending) > Fiat, but assets are realistically more valuable than anything but time. Everything else in that list can lose value and can't be used to produce future value. Most people with a lot of money don't keep it in fiat, or even things like gold. It's almost always in some asset which will continue to make them more money.
Well why not prepare for those days with our taxes? created back ups for everyone?
Like gold,bitcoin and fiat are pretty useless when it comes to a power outage or internet outage, Sure you can use gold to make stuff but you really would be more worried about food water and shelter. Bitcoin you could not really do anything with physically and fiat here in Canada is plastic so you could not even burn it for fire wood. So what is the difference if people adopt a time based currency, If I was the government I would provide watches that hold the network on them so even if the power goes out there is still a wifi network holding the system up, kind of like that red pill free internet thing (in case of a meteor, tsunami, poles changing, sun flare, nukes) something big.
Could they not keep time tokens in exchange for assets that will make them more time as well, if let`s say time does become money?
I'm not sure what you mean by "back ups". If you're talking about disaster relief, that's already paid for with taxes. If you're talking about some kind of technological backup, you'd need to convince the US government to pioneer the project and take almost all of the money out of the welfare expenditures they make and put it towards that kind of project, because it will never get funded outside of that scenario.
As for the infrastructure required to maintain some kind of "time-based" currency, it's going to be no better than the current wifi network we have, essentially. That and fiat can already be considered a version of a time-based currency, which is constantly devalued as more people do jobs. Time-based currencies would have no benefits over what currently exists, and might as well just be called fiat currency with extra steps. That and whatever free-internet concept you're thinking of will probably be destroyed in any of the scenarios you mentioned. The best thing that could be done would be hardening the equipment and waiting for the disaster to blow over. Which would probably mean that trade is starting again, and fiat/gold will likely slowly become accepted (the latter if society was completely thrashed, like a pole change or nuclear war).
If people were smart there would be farmbots all over and backup generators and bunkers/housing for everyone. At least give our species a chance.
Don't worry, those already exist, just not for you.
You could mine all the gold in the ocean and galaxy and we would not have enough to pay it back, instead with this system there is national world wide global gains. The way this currency is distributed. Yes there would be a scarcity for luxury items which people would trade their earned time/duration with (from working) or given time with (for voting or being on the network). People can still enterprise, there is still going to be people even if they are provided food, water and shelter who want the new game or want the new car or phone etc. Want to ride in Elons Rocket to the mars? that will be 5000 hours. Want to buy my peaches from my farm? That will be 10 minutes a peach please. Satoshi, you can have them for free and anyone else in the work force. 
