You need to ensure voltage of the accessory ON switch wire...it's the one that sends the ON/OFF signal to the HU. There should be two power supply wires....One is a constant 12v feed that actually powers the HU, and the other is the 12v that tells the HU when to turn on and off. The later is the one I suspect that might be falling below acceptable voltage while you turn your key on and off.
Properly used butt connectors are almost always good connections, but sometimes w/ the limited space to work with, they are not always crimped tight enough. As an added bit of insurance when working with butt connectors I use dielectric silicone compound squeezed into the connector prior to crimping.
The two power wires I speak of might be colored ( check the manual to be sure) RED- Constant 12v power supply, (EDIT) YELLOW- Accesory on switch.
Again, I am only guessing whats going on, but it needs to be ruled out that voltage does not drop below the proper range ( again it should be listed in the manual as operating voltage range ).
I'd start there with checking that out, maybe someone else has some other things for you.
Good luck, and I cant wait to upgrade my SR5 stereo to a pretty touchscreen!
EDIT- Zoomed in on your wire harness pic, I would think the wire I suspect is the yellow one next to the red....AGAIN just a guess, w/o a manual to check.