Open the box in front of the courier before signing.
Yeah. I don't know any parcel delivery, or even document delivery courier, that'll let you do that. You have to accept the delivery before you can open it.
The only time you're going to get a box open before you sign for it is if set-up is part of the service that's included. For example, "white glove" freight includes bringing the item into the destination, unboxing it, and doing basic set-up functions. Or a local furniture store's in-house delivery.
So it's not just UPS policy that they can't have the package before they sign for it, and they can't open it until they sign for it. USPS, FedEx, Ontrac, Prestige, LaserShip, or any other parcel delivery service or courier service. Telegrams were like that, too.
Don't feel bad about insisting on that signature. And don't let the bullies get to you. They either want the package, or they don't want it. And if there is concealed damage, there are procedures to deal with that, too.