If by "old school" bikes you mean non-indexed shifting bikes, they don't need barrel adjusters because you could "trim" any shifting issues with the downtube shifters.

The reason for having TWO barrel adjusters is this: The barrel on the rear derailleur is generally for dialing in shifting while the bike is in a repair stand, or otherwise not in use. The second barrel adjuster by the handlebars allows you to adjust shifting on-the-fly while riding your bike. That would be mostly in a racing context, especially after having had a rear wheel change and needing to dial-in the shifting while still racing.

Bob