Aren't the a lot of those legacy systems being replaced during the RERP phase? I thought I read somewhere that the landing gears will be replaced along with the APU's and several other legacy systems. The AMP just replaces the avionics, but it must be performed prior to the re-engining phase due to the digital controls in the new engines. Anyway considering all that has changed on the C-5M, I would be concerned too about raising the landing gear on the first flight. Apparently they did later raise them, because there is a picture of it in one of the linked web pages on post #1.
As far as I understand, the only things to be changed at RERP are the motors, and yes, they have to go through AMP first (actually they have to go through EPSU - Emergency Power Supply Upgrade, before AMP, but that's separate,) in order to accommodate the digital controls. Any legacy systems that are changed during EPSU/AMP/RERP are changed as they show up failing. In fact, 84-0059 (the crashed C-5 at Dover) had just undergone AMP at Robins AFB (the C-5 depot since 1999) and while at AMP, it received a mini-depot inspection/overhaul, in which both APUs and all landing gears received extra care.