If operation of the STI shifter IS causing the front mech to move, just not enough to shift the chain either onto the outer ring or the inner ring, does that mean that when you to try to shift to the inner ring, the amount of cable released by the shifter is more than the amount of movement by the mech, and consequently there is excess/slack cable, i.e. you can feel with your hand that the cable is under no tension and is loose? That would indicate the problem is with the mech (or with the movement of cable through the bottom bracket cable guide, e.g. if it were gunged up).
You say the front mech will not move all to way to the stop/limit screws in either direction – is that based purely on using the STI shifter, or have you confirmed this, e.g. by pulling the cable as much as you can by hand to mimic a shift onto the outer ring?
If you had a double chainset, and it refused to shift onto the big ring, that might suggest the cable was not properly clamped to the front mech and was slipping between the clamping surfaces, but for a triple to be stuck on the middle ring and move, but just not move sufficiently either way to change gear, suggests that the problem is with the front mech. I would have a very close look at the mech to see if there were any foreign objects/grit lodged in the mechanism.
NB I presume no one has undertaken any work lately on the bike, since the obvious cause then would be if someone had incorrectly re-clamped the cable (incorrectly routing the cable in a front or rear mech where it is clamped, can sometimes alter the amount the mech moves for a given amount of cable pulled).