Bipolar Transistor (BJts)
Vce is just the common value used for small signal transistors at mA of ce current. Look at pdf's of power transistors (MJE172, etc) and ce is proportional to the current times the internal ce resistance.

The value of ce can be 2 volts or more for transistors with a high ce current.

You are correct that the 'be' voltage is 0.55 to 0.7 volts for silicon bjt's. But that has no direct relation to the ce voltage. You could figure out the base current times the beta of the transistor, which with a grounded emitter would 'predict' the ce current, but the collector voltage, internal resistance and its power rating invalidate such a simple equation if used on power transistors.

Perhaps for small signal bjt's that might hold true, but power transistors and Darlingtons...well, you just have to look at the datasheet for that part. They will give you the hFE but also ce volts at say 10 amps, which maybe 1 to 2 volts, regardless of hFE.

Driving the base with more current lowers the ce of any bjt transistor, but at some point ce resistance starts to dominate and ce voltage rises with more current, until the transistor melts and shorts out.

That is why power transistors often have over current and over temperature protection.