Interesting - I did not read any of this before I started a wholesale LED replacement on my 92' 250 Moho yesterday...but guess why I am here now
I had similar but not identical results as described above. In the Moho I have 12 light fixtures that provide directional and flashing functionality (blinkers). two in the front, six on the sides and four in the back ( I have about 12 other fixtures (head lights and running lights) that only come on with the light switch. The blinkers were a mixture of 1157's (front and rear) and 194 (side markers) - all available as LEDs.
Now, at least in my case, it gets interesting. ALL 12 'blinkers' worked out of the box - i tested left turn, right turn, flashers and all tested fine. That is until I turned on the light switch. Once the lights are turned on the blinking no longer worked - but, note, the 'light' portion did work - you just didn't see any blinking?!?
I took a sample 1157 bulb and 1157 LED to the bench and they both tested fine...I even cycled thru quickly powering them on and off - I checked continuity and on the bulb it was as expected but the LED indicated an open circuit ? I assume there is some circuitry in there and I will disassemble the LED when I have time - but I suspected that this fact might have something to do with the problem.
I swapped all four rear 1157 LED's back to the bulbs...and re-ran my test...and everything started to work again. Even the 2x 1157 LEDs in the front worked properly i.e. they stayed illuminated and you could clearly see the blinking - even though the same LEDs are involved in both ( The bulb 1157 have two separate filaments) ?!?
So that is how I left it yesterday - I need to look into this 'electronic' flasher doohickey - anyone got a part number to suggest.