For a bitcoin exchange, only half of the business is in bitcoin. The other half is in fiat, and I think that's why exchanges need a money transmitter license.
Nah, a business that's 100% bitcoin-based with no fiat on or off-ramps will still need a money transmitter license.
For example an exchange to trade one cryptocurrency for another (bitcoin to litecoin). So even a non-fiat company can indeed be a MSB.
It's completely true to say that the IRS deems it property, not currency, and Fincen deems it a money-like product or a money-substitute. It doesn't outright call it currency though, but the implication is there.