Firstly your new baby does not have a short coat. She has only one coat, and as such she has no undercoat to trap heat next to her body, as the cocker does. It's an important distinction.
I'm certain your Vizsla will love the snow. Mine do, and all the ones I had before them loved snow also. Snow is no big deal to a Vizsla. They are quite adept and agile in the snow, as your cocker will soon find out.
If she is left to just sit outside in the snow, yes she will get cold. If she's moving around and playing and hunting, she'll be fine. Each dog is a little different with respect to it's tolerance to cold. just as humans are.
What will give her problems is combing any of these conditions Cold, Windy, Wet. Individually they are not a big deal. Put them together, and her lack of an undercoat can quickly compromise her.
There is nothing wrong with having a coat or vest for her, and additionally an old towel. She may not need them while she's horsing about, but having them with you while she's romping about in the cold and wet is a very prudent measure. I live in Connecticut
and carry them with me if the dogs are in the woods. I don't leave my dogs unattended outside at any time, so it's not a concern at home.
If "your good 'ol boy" has problem with "dressing her up". Get some Camo gear from Cabela's. Hunt vests, Neoprene Vests, etc. Tell him it's tactical gear for dogs.