It describes something of poor quality, which is, clearly, a concept we all need on a more than daily basis.
So that's the sort of wattle not to be.
As for the sort of wattle not to have, well, this is a bit anthropocentric, but I really don't aspire to one of these:
Icelandic rooster. The wattles are the red dangly bits. Photo by Landnamshaenan
Red wattle pig. Photo by Mark Whitby: https://www.flickr.com/photos/56209874@N07
or either of these:
...or to its human equivalent.
Mind you, given the choice, as we probably will one day, between scraggy wattles and a double chin...
Thing Not To Be Or Have Today: wattle. No one knows where the word for the droopy fleshy things comes from. The Midlands word for something of poor quality is puzzling, too, but might have something to do with the wattle that describes the wooden frames filled with twigs which have been used to make the walls of buildings. This word was watol in Old English and is related to wethel, which means wrap (as in bandage) presumably because you filled in the gaps between the twigs with whatever you had available to stop the draught.