Yup. Spells like Magic Weapon (to choose a spell in the PHB and earlier editions of D&D) have always only targeted physical weapons, not body parts. Unarmed Strike was included in the list of weapons in the first printing of the PHB simply for convenient reference, not to indicate it was now considered a valid target for spells that target weapons. When the realised this had caused confusion for people unfamiliar with earlier editions of D&D they removed it and issued the errata. At this point changing the wording all the way through would have been too costly.
Now, there is no reason 5e couldn't follow 3rd edition and have more feats to support unarmed attacks. In 3rd edition you needed Improved Unarmed Strike (so you didn't get clobbered with an opportunity attack), Weapon Focus: Unarmed Strike, and Weapon Specialisation: Unarmed Strike (Fighter only). I assume 5e didn't include these because they were considered trap options - even with the feats unarmed strike was still (slightly) worse than a dagger.