I never had this problem with watering from above with a fine rose. Three or four passes to let the water soak in between each watering. So I wonder which compost you are using, why it is so dry and why so reluctant to soak the water? Maybe firm it a little so that water is held better. I push my compost pretty firmly into each module.
For the module trays, it may be that those plastic trays do not work well on capillary, as you suggest they have only a small opening at the bottom. But overhead works fine, preferably in the morning so leaves are dry by nightfall, and not every day if it is cloudy and damp. Hope that helps.