Without a mechanical power switch, one would think that the TI engineers have opted for a standby (sleep) mode, and a "soft start" button. Knowing the relatively high current consumption of that era's integrated circuits, a standby mode would drain the batteries in no time.

The single-layer board is in fact a crude switch mode power supply (picture on the left), it provides -3.5 V for the VSS of the integrated circuits (having about 9.5V between their power pins with the +6V of the batteries), and -12V for the speaker and the VFD's segments.

The regulation is performed by two Zener diodes which shuts the oscillations when at least one of the two voltages are reached.
There are two switching transistors (Q3 and Q4) which are wired in parallel and connected to the transformer's primary, another one (Q2) is used to stabilize the voltages via the diodes, and the last one (Q1) allows everything to start up or to let the CPU "kill itself".

By observing the schematic, it is visible that the ON button (top right of the keyboard K2) is at the intersection of VBAT (6V) and the STARTUP line, which is shared between the microcontroller (used as an input to read the keyboard's column), and a pin of the power supply board. By pressing this button, it gives a "kickstart" to the oscillator, which once started, provides the voltage needed to run the rest.

But what happen when the ON button is released ? The STARTUP line would go low, so the power supply should turn off ?
Indeed, and this is where the microcontroller comes into action with the PD (Power Down) line. When the microcontroller starts, one of the first tasks it performs is setting this line high to "keep istelf alive". An internal counter is periodically incremented and reset to zero when a key is pressed. When this counter overflows (after about 2 minutes), the microcontroller puts PD low to cut power and thus "kill hitself."

So finally, there is no soft start, so absolutely no power use when the S&S is off, and the microcontroller is able to shut the toy down automatically if it isn't used (kids can forget to turn off their toys!).

To avoid having to constantly turn back on the S&S to do my tests, but to keep the keyboard column with the ON key functional, I connected the two pins marked in red (PD to VBAT) via a 1k resistor. This forces startup when VBAT exceeds approximately 4.5V, and avoids a short circuit when the microcontroller tries to put PD low. The diode D1 prevents the voltage from going back to the keyboard.