I think you folks are selling Lee Adama short on several levels:
1. Commander Lee Adam has repeatedly shown his independent thinking from his father's several times since the destruction of the colonies. On the day of the attack itself, he defied his father and supported the goals of the new President. When then Commander (now Admiral) Adama ordered the President arrested, Lee Adama refused to carry out the order and was arrested by Colonel Tigh. You better believe the scuttlebutt about those events in the fleet are well known and people know that Lee Adama is not a 'yes man' for his father, but rather has his own mind and is capable of making his own decisions in his own right.
2. As the XO of the Pegasus, even if only for a short time, he showed that he knew what he was doing far better than the Chief Engineer-turned Commander. Apollo knew it was a trap based on the intel pieced together by Starbuck. He told the Commander that, and told that to the Admiral. He was removed from the bridge by the old Commander because he refused to back down from that position. When the three basestars jumped in, he was proven right. You better believe the command crew in the CIC during that event are telling everyone they know on board that Apollo was right all along...
3. Thanks to the efforts of the Commander who never should have left engineering, the Pegasus was able to get out of the mess, but it was their new commander's fighting tactics that kept them alive long enough for the engines to be fixed.
4. When the old Commander left the ship in Apollo's hands, that left a legitimacy to his presence, much like the legitimacy of the Engineer having been a part of the Pegasus's command structure under Cain. Tigh would be seen as nothing but Adama's flunky sitting on their bridge. Apollos is his own man, as he's proven over and over again, and he was there when the dung hit the fan, and he got them out of the jam they were in. That will count for a lot to any crew.
5. Tigh would be the worst possible choice. First of all, you know damn well the fleet hasn't forgotten his martial law at the first part of the season, no one has forgotten his drinking, and his power-hungry wife. Beyond that, he isn't a good Commander, and he knows it, admits it, and reiterates it any chance he gets. He is a good XO, a good right hand man, but he knows the command is not for him and doesn't want it.