Traditionally, the name Gavriel is translated as "Gd is my strength", or more literally, "My strength, Gd".
Gavriel is first mentioned in the book of Daniel, where he speaks to Daniel to let him understand why the Jewish people were not being redeemed at that time. He appeases Daniel by letting him know that even though the people are not being saved now, they will be saved in the future.
From a Zoharic perspective, Gavriel would then be Representative of anything dealing with the sephira of "Gevurah". Meaning, a form of strength which comes from withholding something. As the mishna says, "who is mighty? He who withholds his yetzer hara". This could be understood as being more concerned with delayed gratification rather than instant gratification.
By holding back at the right moment, a longer lasting achievement is made, which would have been destroyed by satisfying the urge at the current moment.
Thus, when the angel Gavriel is evoked in a Midrash, it is usually there to show us a "moral dillema" between what would have been correct and right for that very moment, but which would have resulted in a worse consequence in the long run.
As an example for the midrashim you brought up.
- With the coals, even though Moshe was then worthy to be king, at the time he had to be humble and wait to show his malchut status later.
- In the basket, even though it would have been better for Moshe not to be raised by a non-Jewess, for the sake of the Jewish people, he had to be raised in the palace to be protected from being killed.
- With esther, even though Acheshvrosh was the source of the problems for the Jewish people, by feasting with the vessels of the Beit Hamikdash and thus removing the divine preseence to allow Haman to hurt the Jews, pointing that out at that time, would have lead to the Jewish people being massacred.