I believe the term you're looking for, Peter, is backbudding. Buds that grow closer to the trunk, or further back on a branch, and that eventually form shoots and later on even new branches.
Every type of pine responds differently to a technique. Ryan Neil almost covers them all in those two videos about pines. The audio is horrible, but the information is awesomely complete. I've watched those two videos about 6 times and still I find new information every time. The red line is pretty clear: just hacking away doesn't cut it, things like excess stored energy and vigor have to be in order first.
Sometimes, like in mugo pines, sunlight and good feeding is enough to encourage back budding.
Sometimes, like in scots pines, sunlight and good feeding and cutting fresh shoots in half, will encourage backbudding.
Sometimes, like in JBP, good feeding and cutting the candles encourages backbudding.
Sometimes, strong growth is enough to activate dormant buds everywhere.
Sometimes, like in Eastern White Pine, nothing really works except for cutting the buds off in winter and hoping for the best.
Sometimes, these techniques are combined, there are about as much variations possible as there are pines in the world.
But most of the times, this is a process of at least 2 years. Especially in single flush species. It takes time and patience. I have a notebook that allows me to look back into the past and to see which technique worked for a certain type of pine.
What would be the best technique for your pines? I don't know. It seems like nobody has ever described the process for pinus albicaulis. This means you could be the very first to find out which techniques work and which don't.
I suggest you set up a trial and see whatever works. It could be a good lesson for the bunch of us too!