TLDR: Yes. The USA constitution says the government has power to borrow and lend and to erect buildings where hired people do that. If they combine those powers to erect a building where hired people borrow and lend money, that’s a bank, one that operates on behalf of the national government. Therefore the government has power to create and operate a national bank.
Now here is the longer version.

Background and motivation for researching this

In the USA the government claims to follow a foundational legal document called the constitution. All laws created by the USA government are supposed to be compatible with this constitution. If a USA citizen thinks that one of their country's laws is not "constitutional," they can try to convince a federal judge to publish a ruling that the law is "unconstitutional." If that happens, it means the law is invalid and cannot be enforced by any government agent such as a policeman.
In my experience, government agents actually abide by the decisions of these federal judges. I'm actually not sure why they value the opinion of federal judges so highly, but they really do seem to, at least in my experience. As a result, it can be useful to come up with a convincing argument for why you think a law is unconstitutional, because if you can get a federal judge to agree with you, you might get a law "canceled" -- which is nice because I like the idea of having fewer laws. Also, I just find this stuff personally interesting. So sometimes I spend a bit of my free time analyzing government laws to check if I can make a convincing argument for why they are unconstitutional.
Today I decided to check out the constitutionality of a national bank. This topic appeals to me as a bitcoiner because bitcoin is an alternative to banking and our current banking system is confusing and seems broken, so maybe with a bit of research I could help get it canceled. (Answer: nope, didn't work -- my research convinced me that a national bank is, unfortunately, justified by the text of the constitution.) Here are some of the things I found, just in case anyone is interested.

Constitutional basis

Provisions of Article 1 section 8

Article 1 section 8 of the USA constitution says the government has these powers, among others: "To borrow Money on the credit of the United States;" "to pay [its] Debts," "To coin Money…[and] regulate the Value thereof." It also gives them power “for the Erection of…needful Buildings,” and “To lay and collect…Imposts” – i.e. fees. It implies that they may issue securities (which include bonds) when it gives them this power: "To provide for the Punishment of counterfeiting the Securities and current Coin of the United States." source

Other provisions

Article 1 Section 9 says the government has power to make expenditures from a treasury and report income from prior expenditures: "No Money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of the Receipts and Expenditures of all public Money shall be published from time to time." Article 2 section 2 gives the government power to appoint people to carry out government functions: “[The president] shall have Power…[to] appoint…public Ministers…and all other Officers of the United States.” Article 2 Section 3 says these officers may be paid, which means the government may hire people: “[The president] shall Commission all the Officers of the United States.” source

Application to the borrowing power

The power to borrow money and hold it in a treasury is explicit in the USA constitution. But borrowing money and holding it in a treasury is the same thing as operating a fund that people can put money into and take it out of later. That’s a bank. The government also has explicit power to charge fees for this service, because it has a discretionary power to lay and collect “imposts,” which are fees. The government also has power to pay an interest rate on borrowed money: “The Congress shall have Power…to pay [its] Debts.”
That’s not the only place that gives them power to pay interest rates, it’s also implied in their power to pass laws to spend the treasury’s money (see article 1 section 9, quoted above). That power is a discretionary power to spend whatever money is in the treasury as long as they pass a law saying what they will spend it on. If they pass a law that says they will pay a certain interest rate on money that they borrow, then voila, they have a place where people can deposit money and get it back later with interest – i.e. they have a national bank.

Application to lending power

Now any bank that chooses to pay interest to its depositors must make an income somehow with the money they borrow. They usually do this by lending the money out at a higher interest rate than the one they pay to depositors. The government doesn’t necessarily have to operate this way but it is empowered to, because the power to loan money is, I think, implicit in the aforementioned powers. Loans are a type of expenditure, and the government is explicitly allowed to make whatever expenditures they want as long as they pass a law authorizing the expense.
For example, when the government buys a bond instead of issuing one, they effectively loan money out and expect it back later – usually plus interest. The government is explicitly authorized to make discretionary expenditures, and loans are just one of those – a purchase made now of something to be received later (i.e. a reimbursement plus interest).


Therefore the government is authorized to operate not just a bank, but a pretty standard one, with the ordinary powers of borrowing and lending for profit just as you’d expect a bank to do. As a bitcoiner, I don't really like this conclusion, but there it is. At least I had fun researching it.
Here's the realization that will make your head explode:
The government doesn't create dollars (silver coins) anymore, the bank creates dollar bills (paper). The dollar bills we have aren't minted by a mint, they are bank notes printed by a bank. Some of the notes are printed on junk metal, that is what we call coins. These junk metal debt notes are also created by the bank.
The note says "you can pay X amount of debt at any of our banks with this note", which happens to be all the banks because its centralized.
We have bank notes, not treasury notes. The bank no longer deals in metals, only in debts.
That's why the dollar says "This note is legal tender for all debts public and private".
The dollar is a "Bank Note" that you can legally "tender" for a debt (to pay a loan).
When you agree to purchase something, you technically are creating a debt. You stand there with all your groceries, the clerk scans all of them and says "how would you like to pay your bill" (implicitly). You have taken receipt of the groceries, and now its time to pay for them. When you give the store dollars, you are "tendering" a bank note for this debt.
The constitution intended for the country to use actual money, but it was written before the legal justification for fractional reserve banking was made in England.
Banks are debt mines. Debts are easier to move around than metals, all you need is an agreement (preferably written down, so the judge can have something to look at) and somewhere to write it down (a bank ledger). Then you use the government (judges and sheriffs) to enforce the agreements to settle the debts.
Government is a CORPORATION: The so called "constitution" is no more than a piece of paper used for the corporation as internal rules.
If you position yourself inside that corporation, you MUST obey their rules. If you position yourself outisde that corporation, you just give them a nice fuck off and use Bitcoin as you like.
Once people will understand how to be sovereign and stop bitching around useless "laws", "constitutions", "governments" etc then people will have freedom and truly free market. Until then is just SLAVERY, no matter how you want to wrongfully call it "democracy", it is not freedom.
Solid work, the US Constitution is incredibly dense and flexible. Consider serializing your research work as the US Constitution and Bitcoin are the most important human inventions.
The government can indeed play an important role in the allocation of capital.
The American population has welcomed the US government's role in money creation - through state guarantees on bank credit during the COVID crisis. This move away from Central Bank "money creation", as you say, is constitutional but not limited to loans and deposits. Importantly, this may not be a one-off.
While the American people's ask was a temporary relief / emergency measure, there are signs that direct liquidity injection from the USG will continue, in different forms - such as an emergency war relief package, which is well telegraphed.
Finally, unrelated to your post, USG is also incentivized to do so, due to the high national debt level. You can imagine that's not a US-only phenomenon.