Not to mention unless you print money everyone can't save money at the same time - so the "saving is good / debt is bad" intuition from the household analogy can't hold up mathematically.
In reality, savings and debt are basically two sides of the same coin - one is a promise for future consumption, and the other is a promise to forgo future consumption that balances it out.
One of my favorite simple models is sectoral balances - you can divide US dollar holders into domestic and foreign categories, and then divide up the domestic category into the private sector and the government (leaving 3 categories total).
If you don't print money, the net saving should all add up to zero. If the government is running a surplus, that means the combination of the private and foreign sector is going into debt. If there's a trade deficit, then the combo of the private sector and government are going into debt. Etc.
People can still be against government deficits with such a model, but it's not because everyone should save up at the same time (they literally can't). It's because they want the private sector to go into more debt, and are worried about government debt "crowding out" private sector debt. And more private sector debt isn't always a bad thing - in practice it might mean more housing, factories, and other sorts of investments that debt finances.