So is shutting off the display sufficient?

Oh, forgot to answer this main question.

The way the player works is that the display and the Amp Remote wire are tied together functionally. If the display is on, the amp remote wire is on. If the display is off, the amp remote wire is off. Any time that you want the player not to "pop" through the amplifiers, you must shut off the display so that the amp remote wire goes low and mutes the amps.

Is that what the original player does?

The player does a few more things than just shut off the display. Complete details of everything it does are listed here.