My friend's daughter just had an overbite corrected and did not need one - just braces and elastics, but her daughter's overbite was not severe.
The braces/elastics were able to pull her teeth into the proper position and fix her bite. Her lower jaw wasn't that much smaller than her upper jaw though.
I think (?) the Herbst appliance is for when the lower jaw is smaller or not ligned up with the top jaw severely enough that they need to fix this first (can they tell with impressions?) before doing the braces. I think otherwise the braces won't be effective.
For a third opinion, would your son's dentist know? I'm thinking they may be a good person to ask where they'd have X-rays and have seen a lot of kids/teeth, etc. and sent kids off to orthodontists.
ETA: I agree with the others, it's too bad when you sought the 2nd opinion, you didn't (maybe you did) really question why it was so different than the first one you got. If you can go back and ask now - I would.
Personal experience - We waited till our kid was in high school to do his teeth. Here was our dentist/orthodontist's take on it. Our kid's mouth/jaw was still growing. In the end, it all caught up and then he just needed minimal work.
My sister's kids all started very young (tween) and required twice the work, twice the cost. I don't know if they needed it - it was just the approach they went with.
If your child's lower jaw is simply growing and will catch up - as one orthodontist might think - ask about that. One may be trying to 'hurry' it along. Those are the questions I would ask.