First of all, don’t put your gifts in your checked luggage. Don’t put anything in your checked luggage that if lost or stolen, would break your heart. Which means you’re left with the carry-on option.
You may be able to get your wrapped gifts through airline security, but TSA agents have the right to unwrap your gifts for further inspection if deemed necessary. So wrap your gifts at your destination.
Better yet, use a ground shipping service to send stuff ahead and, equally importantly, to ship items home. Is there any reason to try to fit gifts in already-crowded overhead bins? No, and there’s no reason to spend $25 each way for a suitcase loaded with gifts.
And remember, if you do insist on bringing gifts, you can’t bring any liquids through security, so that leaves out gifts of wine or Champagne. But if you do buy alcohol on the other side of security and know you won’t have to go through a security checkpoint again before you reach your destination, you’re OK.
Personally, I’m still a big fan of shipping ahead and shipping items home while on the road.