Your blackberries are suffering from bramble rust (Phragmidium violaceum). This is a type of leaf fungus.
The upper sides of the leaves on infested bramble plants exhibit dark red spots. The coating that you described as 'pollen' on the underside of the leaves is the yellow-orange to orange-red hymenium of the fungus.
When the leaves are damaged, photosynthesis and sugar formation are impaired, which can disrupt the growth of the affected blackberries and make them less sweet.
How does the fungus make its way onto the plant? The spores overwinter on the fallen leaves and on the green leaves that are left on the plant. In this respect, you took the correct course of action when you gathered and disposed of the leaves. The plant will next become infected in March to April, when the fungal spores begin to germinate in damp, warm weather and the fresh leaves start to grow.
This process will continue until early summer. Incidentally, thorn-free varieties of blackberry are more commonly affected by Phragmidium violaceum than "prickly" varieties. Applying one to three treatments of fungicide in the meantime will help to "clean" the plants. However, as the fungal spores float through the air, the risk of a new infection is always present. In addition to disposing of infected leaves before spores have a chance to form, another factor that can help your plants is an airy microclimate that quickly dries off the blackberry leaves after any precipitation. You can create an airy microclimate by trimming the blackberries correctly, tying them into a fan arrangement, pruning the side shoots as required to promote flower and fruit growth and breaking off surplus side shoots in a timely manner.