Definitely go 16″. A 14″ bike is a bit of a dead end. A good incremental wheel size is 4″ hence 12-16-20-24-26/700C is the route that we have taken.
Bought a Raleigh Kobo16 10 years ago and it’s still in service with friends. Solid, steel, alloy wheels, about £80 new. Anything cheaper will have significant cost cutting – check for caliper brakes, plastic levers and steel wheels. We rode that bike to school down the lanes from age 4 over distances of up to ten miles (it was pretty flat).
As already mentioned, Trek, Giant, Specialized etc have moved to quality (read Aluminum) lighter bikes, V brakes and metal levers. Just check for singlespeed, adjustable brake levers, size-appropriate bars and pedals and ability to fit mudguards and a small waterbottle – the kids love that. I think I also lopped about 1″ off each end of the bars. Fit is more important than weight. Oh and add a stand and make the kid use it, otherwise the bike gets dropped and you will be replacing grips.