most of our cities don't make a profit. They're deeply in debt.
This is an apples and oranges comparison. A "profit" is generally an income statement term equal to revenues over a time period minus expenses over a time period. Debt is a balance sheet concept equal to the total amount owed to creditors without regard to how much is due in principal and interest payments each year.
Most cities in the U.S. do have debt. Most cities in the U.S. are not "deeply in debt" in the sense of having debt service obligations that place a serious strain on the ability of those cities to repay those debts and to meet their other obligations to their citizens.
Most governments in the United States below the federal level have very strict limitations on their ability to borrow funds, and typically operate on close to a balanced budget basis.
Only a handful (e.g. Weld County, Colorado) are debt free, and the policy of having a debt-free government is only desirable in the view of a non-mainstream minority.
A government targeting to pay all of its debts still needs to use devices like insurance companies which privately accumulate reserves, or have a line of credit, to manage short term disconnects between revenues and expenses, often call "reserves" or "rainy day funds" because neither revenues nor expenses are fully predictable.
Our cities need to not be in debt.
Why? Almost all governments have big long term projects that are important to their functioning and since government revenues tend to be fairly steady, not incurring debt drastically reduces the capacity of a government to make necessary long term investments.
A government that only buys what it can afford from current revenues without saving or borrowing is almost surely underinvesting in needed infrastructure.
But also, do they need to make a profit? What if a city, or any
governmental entity for that matter, was able to implement a policy
that allowed them to exactly recoup their losses for the year by
taxing just enough to break even?
The problem is that revenues and expenses aren't that predictable (and even when they are predictable are "lumpy"), and the downside to not paying a legitimately owed debt is greater than the downside to having financial reserves in the event that revenues are low or expenses are higher than anticipated.
Keep in mind that the "profit" from excess revenue is going anyway but to pay future expenses. It is profit that inures to private individual owners of an enterprise which makes an enterprise "for profit" instead of "non-profit".