Is it normal for an old sub amp to shut off after a few songs?

Sounds like the thermal protection is actually doing its job, not failing. When an amp gets too hot it shuts down, cools off, then restarts once it's back under the limit. That cut-out-and-come-back pattern is classic thermal shutdown, so don't assume the protection circuit is toast from age.

Dropping the gain will absolutely help. At three quarters of the way up you're likely pushing the amp into clipping on bass hits, and a clipped signal is way harder on the amp than a clean one. Here's the thing most people miss: more gain doesn't mean more output once you're clipping, it just means more heat and more distortion. Turn it down until the bass sounds clean at your usual listening volume and you might be surprised it still plays plenty loud. The old 90s amps were often rated conservatively too, so they had real headroom back then, but 30 year old capacitors and tired output stages don't handle abuse like they used to.

Also worth checking your wiring while you're in there. If those 10s are wired to a lower impedance than the amp is stable at, that alone will cook it fast. Grab the model number off the amp and look up its minimum stable load, then count up how your subs are wired. A lot of guys in the 90s ran subs in ways that made the amp work way harder than it was designed to. On the fans, skip the hunt for the original accessory fan. Those were a band-aid back then too. A small 12v computer fan or one of the cheap amp cooling fans from any car audio shop, wired to a switched or remote turn-on lead so it only runs when the system's on, does the same job for pocket change. Just make sure air actually moves across the heatsink fins and not just around the trunk.

And yeah, keep it out of direct sun through the back window if you can, trunk temps in summer are brutal on old electrolytics. Should survive plenty of cruises with a lower gain and some airflow.

Agree on the cheap fan route. I had an old Rockford punch amp that did the same heat cycling thing years back, zip-tied a little 12v fan right on top of the heatsink so it blew across the fins, tapped it into the remote wire, problem gone. Only other thing I'd add is check the mounting surface, those old amps need solid metal to metal contact to dump heat into the body of the car. Mine was bolted to a piece of MDF like an idiot and that alone made a difference once I moved it.

Before you blame heat completley, check your power and ground connections. I chased a similar cutout on an old amp for months thinking it was thermal, then found the ground was barely hanging on under a painted bolt hole. Amp would pull hard on bass hits, drop voltage, and shut down till things settled. Sand the spot down to bare metal and hit it with some dielectric grease if you got any. Also wiggle test the fuse holder while the systems playing, those old fuse taps get crusty after 30 years and cause the exact same symptom.

Update: turned the gain down to about half and added a little fan blowing across it like the zip-tie trick. It ran a whole hour without shutting off, so the protection circuit is fine after all. But now I'm wondering about that ground thing, mine's bolted down over some old paint too. If I sand it down to bare metal, will that alone fix a weak ground or should I run a fresh strap back to the battery?