If you still want a bit of advice about making your wife feel less exposed here's mine-

I loved breasfeeding but I am not okay with doing it in public (my weirdness, I hate wearing swimmers at the beach too) I could always find clean, comfortable parents rooms at shopping centres or airports etc. The only time I couldn't find somewhere was on planes and buses, I never wore breastfeeding clothes, I usually wear t-shirts so I just lifted enough to feed from underneath, baby's face was covered with the t-shirt and the only thing you could ever see was the side of my tummy.

It's okay to feel weird about breastfeeding in public because you are a bit naked just don't let baby go hungry!