I may be looking at something different:
The data latch for the displays is driven by signal _STROBE_ which is
active when MRD, N2/LOAD, and TPB are asserted. Just like on an Elf.
N2/LOAD is a diode OR of the N2 signal from the CDP1802 and the LOAD
signal from the switch. If it works while in LOAD mode but not with an
OUT instruction, the fault lies with N2. A poor connection, bad diode, etc.
It is really simple so you should be able to make it work. I managed to
get an Elf working way back in the 70's when the sum total of my test
equipment was a VOM and a cobbled together logic probe. And I didn't
have a PCB.
IDIOT is kind of odd in that it uses 7 bit data. Depending on the
first character it sees, it will either echo received data or not. But
the echo is bit by bit which means that it could echo what you send it
just fine but the data it sends looks wonky.
Oh, IDLE mode. If you look at the data sheet you will find that in an
IDLE cycle, CPU puts R0-1 on the address bus and asserts MRD. (With a
note of "Suppress TPA, wait for DMA") So that constant cycling is normal.