Will Alpine Currents regrow if you cut them back to ground level
If I can answer your question this way; Yes you can prune the canes down to the ground, but the question remains is this good for your plant to do all at once, the answer is a resounding no. Pruning any shrub down all at once places it in distress, and weakens the plant.
The better practice is to take a portion of the older growth out each year to the ground, and try to avoid taking more than one third of the shrub in one given year. So if you get into a practice of pruning back older canes on a rotation, you will maintain a healthy balance of newer branches and older ones.
If you also want to manage the height you can also do a peripheral shear to limit height , while also thinning out older canes.