There's no way that's true. Every time I had a cold brew I felt like shit immediately. I immediately get a stomach ache upon drinking the stuff. That's why even if the weather is at 38-40 °C I still drink a warm brew but accompanied with a glass of cold water.
As far as recommending you cheap brands, I just buy one of those jars of instant coffee that they sell everywhere. They tend to be cheap and they last me a long time.