I don't think i understand your recommendation here, how do you cut and burn the private key of your paper wallet, how to you then recover your paper wallet or spend from it, when you do not have the associated private key to the wallet.
The below quote explains it.
But you will need a full paper with a public address, QR code and private key too. If your approach is against private key information, for public use I agree, you can have two papers for two different purposes, one for back up, and one for bringing it around for using in public places.
If you want to use only the public address, the QR code in public places and want to hide the private key at home, you can print two paper pieces, one is full details with private key, public address, QR code that will be stored at home; another paper is only with public address and QR code.
It is not even recommended to carry your paper wallet along with you to public places, especially when the wallet address is holding a large amount of your coins, it is too risky in my opinion.
Is it what I advised, too?
I want to say one thing, the wallet you want to bring around, even just with a public address, needs to be a small wallet. You don't want to leak your main wallet's public address too much.