Is it okay to run a different power amp chip if the pinout matches?

Matching pin layout gets you maybe halfway there. Those little amp chips all look the same from above, but the gain is often set inside the chip, and the pins that control it vary wildly. One part might have the gain fixed and a spare pin used for a bass-boost cap, while another expects resistors between two pins to set the volume of the whole stage. Drop the wrong one in and you get silence, nasty distortion, or a high-pitched squeal, even though every leg landed where the drawing says it should. What I'd do is pull up the datasheet for whatever you found in the drawer and compare three things against the chip the schematic calls for. First, which pins set the gain and whether they're expecting caps or resistors or nothing. Second, whether the output drives a speaker straight or needs a cap in series. Third, the minimum supply voltage, since these kits usually run off a battery pack. If the substitute needs external parts where the original had none, forget it, because cutting into a hand-stitched board to squeeze in a resistor network is misery.

The good news is that chips designed for exactly this job, low voltage battery power straight into a small speaker, are cheap and everywhere. Given how much time you've already sunk into the stitching and gluing, ordering the right part for a couple of bucks beats gambling the final hour on a maybe. The discrete output stage is real and I've done it on perfboard, but it's a pair of output transistors, a couple of diodes to bias them, and a driver stage, and getting the idle current right so it neither cooks the transistors nor buzzes is fiddly. Long hand-stitched runs tend to pick up hum too. Not the project I'd learn it on. If you do try a swap anyway, keep the leads long enough to back out, listen at low volume first, and if it squeals, a small cap right across the supply pins usually calms it down.

On my paper build I soldered a machined pin socket onto the foil traces instead of wiring the chip in direct, and it saved me more than once. I coud try three diferent amps from the junk bin in an afternoon, and when one squealed I just lifted it out without disturbing all that glue and thread. Stitched traces are fragile in a way breadboards aren't, so anything that lets you back out of a bad chip without re-doing the board is worth the ten minutes it takes to wire the socket. Whatever you drop in there, give it its own lead length so the socket sits flat and the paper doesn't take the strain.

Been down this road with a hand-built board about twenty years back, back when we drew circuits with a conductive ink pen becuase etching tanks were too much hassle in an apartment. One thing I'd check before anything else: what current does that output chip actually pull? Some of the little amp parts in the junk drawer will run fine but drink the battery flat in an afternoon, and on a stitched board you don't want to be reworking the power rails later. I ended up gutting a dead portable radio for its amp section and just lifted the whole little board out, wired it in as a module. Ugly but it worked for years.