Could Maester Aemon become Hand of the King?
Let's say that Daeron didn't die, and became king after Maekar. He's still being tormented by his dreams, which makes him not very capable of ruling, and he knows it, which is why he would choose a Hand who he completely trusts and who wouldn't try to take power for himself and his family, so the choice is between his brothers or Bloodraven, but I think if Brynden really is scheming against the royal family, Daeron's dreams would clue him in enough not to trust him. That leaves Egg and Aemon. Aegon is probably already unpopular with the lords, and he has a family and presumably Summerhall to take care of. This, and the fact that Aemon has served him as maester on Dragonstone for more than a decade at this point, is likely to make hin Daeron's first choice.
Would Aemon accept it? He didn't go to King's Landing when his father was king so he couldn't displace the Grand Maester, but this situation might be different enough for him to make a differrent choice. Would the lords be bothered by a maester kinda going against his vows? (Technically the sacred vows are about not holding land or lordship and being celibate, which wouldn't be broken.) In canon, some lords wanted to make Aemon king, so they might not raise much of a stink, but of course the ambitious ones who want to become Hand themselves might protest.