Britain did have economic sanctions on Iran in effect. The entire oil industry was boycotting Iranian oil, and the economy was quickly going into crisis. This was leading to all sorts of political problems which the US thought would play into the hands of the Iranian Communists (Tudeh). And that was the reason the US backed the coup.
The problem is that Mossadeq by that time was not willing to compromise. He had defeated Anglo-Iranian,a nd they had come back to him with a deal that Iran would have quickly accepted at the beginning of the crisis. But Mossadeq wasn't willing to back down an inch on the nationalization even though it lead to the boycott which was ruining Iran's economy.
Mossadeq was a very strange man. Not quite the democratic hero of legend. He seems to have been a truly decent person, but by the time of the coup this "democratic" leader was ruling by decree and the Iranian parliament (majlis) had totally surrended their power to him. He seemed incapable to making political comprises which is at the hear of democracy. He was indifferent to the collapse of the Iranian economy because of his stubbornness. The US was originally very pro-Mossadeq, but the more they dealt with him, the more alienated they became.
There is no guarantee that Iran would have developed into a strong democracy under Mossadeq. He might have become a fairly benign dictator - but one can argue that the Shah was so for most of his regime. We don't know how Mossadeq would have changed after his government became flush with oil wealth, especially in the 1970s. Would he have remained a personally decent person, or adopted harsher or megamaniacal tendencies by being corrupted by the oil wealth?
It's hard to see a good resolution to the crisis with Mossadeq's personality. One could see a failed coup attempt which was very farsical at times. But Mossadeq agreeing to a compromise? I don't see it untless you just want to handwave it as part of a thought experiment.