I very much doubt whether a BEC is in the ESC. Simplest way to check is to use a voltmeter and measure the voltage at the receiver pins. If it 5 volts, a BEC is involved, if 7.2 volts is shown, there isn't. The receiver might have an internal BEC to allow itself to work on the higher voltage, but that is not something that I would consider trusting.
Going off the vintage, it probably uses and suffers from the same transistors as the Hitec Gold Brick. The voltage limitation there was due to the power that the resistors driving the output transistors needed to dissipate. All of the ones that were shown to me had big black burn marks where the resistors used to be, usually caused by running the thing on its highest rated voltage. Or more..
There is probably a very good case for doing a comprehensive refit involving radio, ESCs and servos, much like hat happens with real warships.
To scale, I would expect that on 6 volts, under load, the motors should be pulling about 10 Amps. If the basic power is not there, the performance won't be, either. Back in the elder days, just having the model go was a triumph, having it go like it should and under control was luxury. Now, it is expected.