If it is so, everything falls into place
Sorry no. when there is lots of current there is only a small voltage across the transistor. Look at the data sheet for the exact value it is known as Vsat of Vce. However it is this small value that is giving you the few watts that istoo much for this transistor without a heat sink.
The point is that when the transistor is off there is no current flowing so zero times nine volts is zero. That point allows the transistor to cool a bit.
If you use a FET there is much less voltage dropped across that when it is on and it runs cooler for a given current.
The DVM (a cheap one) will only show normally an average voltage and not very well at that.