It makes a lot more sense to buy a used bike and then sell it off again when your son outgrows the bike. This works best if he is capable of simple repairs and adjustments which older bikes may require. I've done this for years to acquire decent bikes kids in my scout troop can have when they attempt the cycling merit badge. I do the maintenance and adjustments before turning the bikes over to the kids. If you look for a used road bike you should look for one with a 23" frame. On a mountain bike the frame size should be about 1 and 1/2 inches shorter. I looked at the Minneapolis CL to see if there were any decent bikes that would fit your son. They are not all junk but you probably will have to settle for an older bike. Here are four that look like they would be the correct size and are easily within your budget plus allow extra money for things like replacing tires and brake pads (often necessary on older bikes). You are correct that there are quite a few overpriced bikes though. People look for a brand name like Schwinn and pay way too much when there are many good brands to look for.
Trek 930 singletrack - $120 (inver grove hts) This could date back as far as the mid 1990s and sold for around $500 brand new in 1995. It would have been at the upper end of entry level bikes. If it is in good working order it would be worth the asking price ($120) even if it needs new tires and brake pads. It should be the correct size for someone who is 5' 7".
https://minneapolis.craigslist.org/dak/bik/d/inver-grove-heights-trek-930-singletrack/7312884652.html
Specialized Sirrus 23" Road Bike - $250 (West Side of St Paul)
https://minneapolis.craigslist.org/r...308959536.html $250 23” frame which is about right for your son's height
Fuji Allegro road bike - $180 (Inver Grove Heights) This is an older road bike with 27” tires so it could date back to the late 1970s from the components listed. It would be better than any entry level bike you can buy new these days.
https://minneapolis.craigslist.org/d...310670476.html
'
1990s GARY FISHER Aquila 21 speed frt susp MTN bike 50cm - $120 (Plymouth) This one could be either 1993 or 1995. I'm not thrilled with mountain bikes that have front suspensions unless they are really high end but at the price shown here it would be worth looking at if it is nearby.
https://minneapolis.craigslist.org/h...301898760.html