Glad to hear your carb swap solved the lost idle circuit issue. That said it still sounds like you’d like to resolve the issue with the non-idling carb.
You’ve indicated that the idle circuit passages all appear clear, i.e., either air or fluid transfers through all of them. However, the results of having no idle circuit suggest otherwise – either the idle circuit has a huge air leak and you can’t supply enough fuel via the metering needle to make it run, or there simply is no fuel entering the idle circuit. I am not familiar with the specifics of your particular Linkert, but they are all similar enough that the following thoughts might help you with the next stage of diagnosis.
Using just the carb body it’s pretty straight forward to troubleshoot the idle circuit flow. The images below are of a model M53A1, but should be close enough to your model that you can adapt accordingly. I like to trace the idle circuit by applying a gentle stream of air to it, and monitor the outflow by submersing the carb stem or entire body in weak soap water. To perform this test you need to plug (with clay) all the drillings in the main stem except the bottom outlet. The power needle and/or main jet need to be closed/plugged to shut off those outlets. If the carb has a direct connection of the air correction circuit to the main well this likewise needs to be closed off. The hole at the top of the main well where the main nozzle enters the carb body also has to be closed off with clay. Once these openings are all sealed you simply apply light air pressure somewhere on the idle circuit and watch where the air escapes.
The internal idle circuit connections you are especially interested in probing are the drilling from the main well to the idle mixture metering seat (primary supply of fuel to the idle circuit) and from the idle mixture seat to the greater idle mixture galley, all near the bottom of the stem. Hopefully the image below (courtesy of Tom Cotton) will allow you to visualize the foregoing description.

- Sectioned stem.jpg (69.95 KiB) Viewed 823 times
The image below shows how you might test for airflow through the idle circuit. For example, when the idle circuit is lightly pressurized, and no idle mixture needle is present, you should see air flowing freely out the bottom of the stem, which would indicate that the fuel supply from the main stem is wide open. If you now introduce the idle mixture adjusting screw and begin tightening it down, the bubble rate at the stem bottom should begin slowing and totally cease when the needle is fully seated. When all the orifices are allowing air leakage, covering any one of them (the idle slot at the butterfly, the idle mixture needle hole, the bottom of the stem, etc) increased leakage at the other orifices should be observed.

- carb with upper main well sealed for testing.jpg (59.89 KiB) Viewed 823 times
It might be most diagnostic to perform such tests on both the functioning and the problematic carb bodies to observe if different behaviors exist.
Interested to hear the rest of the story.