One of the reasons a map will not have the little blue icon to start the data collection process is that Collector thinks that there are no editable layers in the map. Validate what you think your editable layers are in that map. You have to have at least one editable layer in each map in order to data collect in the map. I have seen this in my own maps where I forgot to include the editable layer. In the older Collector versions, you would not even see a map in you map list if it did not have a valid edit layer. Field Maps does not operate that way, its shows all your valid maps, but no blue icon if no editable layer in the map.
This might not be your issue, just one that I had seen with a similar behavior. Something to consider. Hope that helps.
