Well, no.
10,000 men traversing the paths requires ten thousand trained as such (the training necessary being less than climbing cliffs, though), each carrying his essentials - that is, probably food for a few days (water may be found), ammunition, his rifle, specialized equipment and so on. All that slows them down, especially in mountaineous terrain, and especially if they can only move across certain paths. Not all of them can move at once, as most ways will be narrow, and so on. In effect, you may have ten thousand men moving slowly as a ten times a thousand men column. That's no fast way to cross a mountain, especially if you need to fight afterwards.
Like evilcherry, I see those mountains similarly to the dolomites - see, for example, this
this picture. This shows all of the terrains - the pass (street in the right half), the infantry-passable terrain (both the part without snow and the snow in the upper left corner), and the cliffs (which I probably won't have to show you ;-) ).
A cheap defensive option would be to build a pillbox on the shadowed peak to the left, with a cannon and machine guns towards the pass and a machine gun to the back. You cover the pass and even with infantry scaling the terrain, you can defend yourself pretty well. Using the view-point for another one, you can also cover yourself versus people scaling the cliff to get to your pass-sided defence. Basically, you block the passes as that's where the attack that can actually hurt you comes from.
Of course, that's just cheap - you can get more and more and more if you want more defensive capabilities. For example, you could build hundreds of bunkers in the cliffs, connected by tunnels. Somewhen, you'll be very, very cost-ineffective.