Where is 650mA coming from?
Datasheet provided clearly shows Absolute Max as 600mA.
Further it shows that you need 50mA of base current to saturate to allow 500mA thru.
Yes, Voltage available to the LEDs will be 5V less the Vce-sat of up to 1V. If you look at the charts that complete the datasheet, there will be curves showing the voltage at different currents.
MOSFETs are better in that regard - a good low Rds MOSFET will be in the 10mOhm range (.01 ohm) and the voltage drop will be small - with 1A of current thr the MOSFET, the voltage will be V = IR = 1A * .01Ohm = .01V.