The US has had a number of dollar coins, but they've never found a lot of use.
The last effort made the coin almost exactly the same size as a quarter (or nickel) and the only way to tell them apart was that the dollar had "flat" edges and an octagonal(?) outline instead of round.
During the few years when people were "trying them out" I probably got more than a half dozen in change from people who thought they were giving me a quarter. There was enough difference that I was able to spot the error to correct it in most cases, but a couple slipped through so I never knew who to return the six-bit difference to.
IFF a dollar bill could only be used ONCE, the cost of printing one would need to be much less than the value. Since dollar bills change hands much more frequently than other denominations, the few months that a "One" is in circulation isn't really much of a problem. The "dollar exchanges" during its lifetime by a particular "One" is about the same as for any other of the common denominations.
The manufacturing cost versus face value is a completely unrealistic way to decide whether the particular specie is useful enough to be continued in use, but the comparison continues to be made by those with no comprehension of economics. (And by some who like to do it just for fun.)
If the printing office got a penny every time a dollar changes hands, a single dollar bill would get them a couple of hundred bucks back during its lifetime. The printer doesn't get it back, but in most states the 8% to 14% "sales tax" on each dollar exchanged is a pretty substantial contribution to the (mis)government. Anything that facilitates exchanges of dollars is of benefit to the bandits government, and the One Dollar Bill is a great facilitator.