My understanding was that you needed the full blockchain for maximum integrity.
You don't. The block files are pretty useless after you've verified them and have gotten the chainstate and other essential information. If your computer can verify its legitmacy, there is no problem throwing them away. You could have problems when importing addresses, etc. None of the potential problems actually affect the security; could just cause some inconvenience.
What if everyone pruned their blockchain, would that mean there wouldn't be any full copies of the blockchain?! Wouldn't that be a problem?
Yes. If everyone pruned their blockchain, then it would be a huge problem. Since the wallet prune itself as you get more blocks, no one would ever see the entire history if that were to happen. And yes, it would compromise the security greatly in that case. IMO, pruning is just for people who wants to run Bitcoin Core and to support the network, not really that much as to helping the network (debatable but thats my opinion).