Grey tinge round mouth = old wives tale (apols to old wives :) ) How would air in the stomach cause colouration of the skin on the face?
If babies need to get rid of wind, most of them will do so, without any patting or jiggling - just sitting them upright often helps them, but this happens naturally anyway with normal handling and cuddling.
Many mothers are quite convinced their babies have a problem with wind - I expect some of them are right, but if a new baby is crying and uncomfortable, it's probably hunger/thirst/need for closeness and comfort (all of which are indistinguishable to a baby and all of which can be satisfied at the breast).
Babies often burp after a bout of crying. The burp is there as a result of the crying, not because of the crying - they take in air when yelling. This is not the other person managing to bring up the baby's wind - it's just the burp came up when they happened to be holding them.
For most mothers and babies, it is a waste of time and a source of anxiety, all that patting and worrying and wondering if the burp was big enough...just keep the baby close after feeds, and snuggled in, and rub his back if you like, 'cos it's quite nice for both of you....:)