You are completely confused. Read the article again.
Fiat money ≠ Private money
When I mentioned BitPay in my previous post I was trying to illustrate that the number of merchants that know about Bitcoin, and are prepared to accept Bitcoin is growing. When there are enough manufactures, wholesalers and retailers that accept Bitcoin, there will be no need to convert Bitcoin into fiat money.
Yeah, I would like a situation like that, however you forgot one thing: diversity. What makes you think that a considerable number of business in many sectors of different economies will only choose BTC? Why not LTC? Or even DOGE? (ROFL!)
BTC is not the only "private money" available in the list of options that all business have. They could even do they own "private money".