Can anybody identify this component (SPRAGUE 663044-4) on this vintage electronic?

Those are small PCBs with a passive filter network in a single inline pin package potted with epoxy. The schematic on the far left within the dashed lines shows the components of a single board (notice, the pin-out for that part of the circuit is labeled 1-7), and it is not drawn for the others (presumably identical) to save space. You can replace them with what is inside the dashed line on the left side of the schematic, but it is unlikely to be what is broken, since everything inside is a passive (just caps and resistors).

Any jellybean NPN transistor (2N2222, 2N3904, etc.) should work fine in this circuit.

The manual is quite detailed in describing exactly what each circuit does. As Zuofu said, the 7-legged orange squares contain the passive circuitry shown inside the dashed line in the schematic diagram excerpt you show. One of these and a pair of transistors constitute one of the six flip-flops in the 48-step counter.

If you're seeing the tempo light, then much of the logic part of the unit is working, so you need to focus on the parts that generate the sounds and buffer them to the output jack. Since you're getting no sound at all, start at the end and work your way backward into the circuit.

Do you have an oscilloscope? It would be extremely useful to have one to debug this unit. Even a cheap soundcard-based one would do for this project.