The government can't support the people, it's always the other way around. The money the government spends, be it from taxation or inflation, comes from the private sector. There's no other source. Without the huge increase in government spending that has taken place in the U.S. over the last decades, people would have the savings to deal with a crisis.
Of course you're right. Productivity or GDP comes mainly from the private sector. We turn over responsibility for creating money to the government however, because we don't trust the private sector to do that. When we tried it, we eventually got into a big mess. So we made the government the source of money, with the private sector being the source of most of its value.
Savings are an interesting thing because when it is determined that the people should save more, the government must run a deficit to provide the additional money for increased savings. If you think about this a while you will see that this is correct. It can be understood this way: The government spends money into the economy and removes it via taxation, or temporarily via bond sales. When the government sells bonds, money that was in the private sector is transferred to the central bank and a treasury liability, i.e, the bond sold, replaces the money that was in the private sector. If the government budget is perfectly balanced the government will remove from the economy, via taxation, exactly what it spends into the economy. There will be no net change in the amount of outside money in the economy.
On the other hand, if it is decided that the people should save more the government must spend more into the economy than it removes, it must run a deficit to do that. If it later happens to sell bonds to the private sector in an amount exactly equal to the amount of the deficit, those bonds, once in the private sector, become the vehicle for additional savings. The bonds serve as an interest bearing store of money. Thus deficits are intimately related to increased private sector savings.
Let us recognize that a lot of the recent large deficits were in response first to a war, then to the threat of financial collapse and a depression, and recently in response to a pandemic.
Even so, apparently, by the foregoing reasoning, it isn't exactly true that.
"Without the huge increase in government spending that has taken place in the U.S. over the last decades, people would have the savings to deal with a crisis."
It is a very hard thing to accept that as populations an productivity grow, or in the event of a crisis, the government must put additional money into the economy by spending more into the economy than it removes via taxation. Thus it must run deficits -- maybe not every fiscal year, but nevertheless quite often. If it doesn't do this, the economy will be thrown into recession and/or deflation, exactly as would happen were the government to start running surpluses year after year.
This way of thinking is foreign to most of us because we have been wrongly taught that government should run just like a private sector business and manage its finances just like you and I manage ours. But, of course, on close examination of the role required of government as the source of our money, this is impossible.
That said, there is certainly plenty of room for debate on the proper size of deficits, the role of government, and spending priorities.