Only thing unplugging the ECU proves is that it's probably faulty if there is no change.
According to the Toyota troubleshooting guide I have for the A340F this is not true. If there is no DTC and basic settings are OK you perform a "manual shifting test" which comprises disconnecting the solenoids and inspecting manual driving operation. Overdrive should work with disconnected solenoids in D position. I assume the difference between A340F and A343F to be minimal.
Since on my car the O/D doesn't engage during this test my problem is a faulty transmission. As I wrote in my original post, I already repaired the ECU.
Looks like it is difficult to find somebody who is familiar with those gearboxes...