Yeah, as others have mentioned- a Sora triple FD should shift just fine. I have a Sora triple FD on a steel frame triple with a square taper crank(all the same as yours) and it shifts fine.
Friction shifting means it doesnt need to match the number of cogs in back in terms of 'speeds'.


If the chain falls off to the outside, then the limit screw needs to be adjusted to limit how far the derailleur can move. This is a simple adjustment if you read ParkTool's website or watch a couple youtube videos. A bike shop should have been able to correct this.

As for not being able to shift into the big ring at all, a couple of blind suggestions are-
- to make sure shifting isnt happening when the chain is on a large cog. Shifting into the large chainring should only happen if you are in the smallest 5 cogs or so. Thats a general guideline, but if its on the largest 3 cogs, then instead of shifting to the larger chainring, just shift to a smaller cog for more speed. When the chain is on a large cog, its more difficult to then pull the chain up to the largest chainring.
- Perhaps the cable is too slack. I would hope a bike shop would notice this, but if the front derailleur cable is too loose, then part of the shifter's movement is wasted with tightening the cable. Once its tight enough, then the shifter starts pulling the cable and therefore the derailleur. The problem is the shifter runs out of room for movement and the derailleur is only halfway or 3/4 of the way to fulling shifting. Hope that explanation makes sense. Basically, you want the cable as tight as possible when the shifter is relaxed(pointing down) and the derailleur is relaxed(closest to the seat tube). This will ensure full possible movement.