Am I failing to hear a subtle difference between אושר and עושר, or do the Hebrew words for wealth and happiness really sound exactly the same?
It sounds the exactly same by most of the speakers. Most people don't distinguish between א and ע anymore.
The similarity between the pronunciations (these days it's totally identical) is used a lot. For example, "And they lived happily ever after" in Hebrew is "והם חיו באושר ועושר עד עצם היום הזה", literally "and they lived in happiness and wealth until this very day".
I have been moderatly rich and I have been moderatly poor. I prefer moderatly rich, cause then you have more freedom to decide how to spend your time and how to take care of youself and others. Freedom to choose is good for health, and good health can make you happy. It all connects. As a famous soulsinger sings: "No body knows you when you're down and out...." Weath is connected to happyness, no doubt in my mind anylonger about that.