The problem with your analogy is that the US doesn’t have a trade deficit with just one country. It has a trade deficit with everyone, which creates a current account deficit.
In your analogy, you have trade surplus with your employer.
But what is the mechanism which causes the Federal Government to go into debt when private companies and citizens import more than they export?
If I import more bananas from some Pacific Island country than they buy of American goods, how does that cause the government to borrow money and go into debt?
The trade deficit is not the main cause of Washington's borrowing. The deficit causes trillions of dollars to accumulate in the hands of Chinese manufacturers, much of which eventually ends up in the hands of the Chinese government (a sovereign wealth fund). The Chinese government keeps that money in dollars (rather than convert it to Chinese currency) because converting it would reduce the strength of the dollar relative to the Chinese currency, making Chinese imports more expensive to Americans, which Beijing does not want. Someone holding $trillions is motivated to invest it to avoid its being eaten away slowly by inflation, and in general, it is not easy to invest that much money especially when you are not located in the US, but US Treasury bonds are one easy way to invest a lot of money.
Accumulating $trillions in US Treasury bonds was never one of Beijing's goals: it is the side effect of Beijing's being eager to help its export industries, which is does to give as many Chinese as possible hope for the future, so they don't revolt.
It’s not public debt, it’s private debt. It’s not a coincidence that Americans are becoming more indebted at the same time as the country is running current account deficits.
In your analogy, you have trade surplus with your employer.