I've been in your situation, and I would also go for an economical BMX bike. All three of my boys, age 6, 8, and 10, are on 20" wheel BMX bikes which were Wal-Mart or Amazon brands, bought used, and they serve them fine. I do my own maintenance but there's not much to a single-speed bike.
The less complicated it is, the less the manufacturer has to cut corners to keep the price down. For BMX I choose regular spoke wheels, V-brakes, and the avoid the gyro-brake adapter (that lets the handlebars spin without twisting the cables). Yes, the bikes are sometimes hastily assembled, but be particular in the store and you'll find some good ones, or adjust them yourself at home.