Princhester:
Using just the big chain ring (at the front) and your son getting low ratios by going to the largest rear cog while still on the largest front chainring is a bad idea because the chain is then cutting across at an angle and will wear out fast. I suppose you might not care about this if the bike isn’t going to last long anyway.
Either the stop screw setting the outer limit for the front derailleur cage is in too far, or the cage has been knocked and isn’t where it should be or there isn’t enough cable tension to pull the cage out enough. The first option isn’t likely because the stop screws don’t usually move, assuming it was set right in the first place.
The second option is a possibilty, particularly with a boy who’s hard on his bike (like I was at his age, no criticism intended ;)).
Did I mention the rear wheel is a little bent also? No? The rear wheel is a little bent. Won’t be able to pass this one down to his brother when he outgrows it.
Having slipped around a little is a possibility. The front derailleur moves enough. With the chain on the middle gear, trying to shift to the larger gear, the chain seems to get pinched between the derailleur and the largest gear, rather than riding up onto the larger gear. I was more concerned with the back gears, and didn’t look into it much, or look at a working bike to see what is supposed to happen.
You can tell if the cable hasn’t enough tension to do it’s job this way: hold the rear wheel off the ground and get your son to spin the pedals. Set the rear gears so the chain is on about the middle of the cluster. Now try to use the gear changer to shift the chain onto the large chain ring. From what you are saying, it won’t make it. Now reach in and push the derailleur cage out a little with your finger while trying not to whack/pinch/cut off your finger in the chain etc. If the cage will move outward and change up, then the stop isn’t set in the wrong place and all you are lacking is enough cable tension to do to the cage what you just did with your finger. You now just need to take up a bit of slack in the cable. They stretch a little.
Just use the gear shift to put the cage in the inmost position (which will make the cable slack) then undo the cable holding screw, pull a little (half an inch?) through and re-tighten. Wait. Tell your son to stop pedalling first.
ETA: I’ll look at it some more tomorrow.
If you mean warped, it probably just needs spoke adjustment to pull it back into line.