Overheating

Go to the local auto parts shop and borrow a cooling system pressure tester. Basically, it is an air pump that you hook up to where the radiator cap goes and you pump it up and see where the coolant comes out. If it has an external leak, you will see it then. Here is a how to guide on running that test: https://www.2carpros.com/articles/radiator-pressure-test

Pump the system up to about 15 psi, not more because you could damage something by putting too much pressure on it. Then you just look around and listen for a whistling sound.

Do that, take a few pictures of the offending part, and post them here. We will look and identify the problem. If you cannot find a leak, then we will discuss the other possibilities.

Seems like you have a weak water pump that is not providing sufficient coolant flow unless you rev the engine a little.

I tend to agree with KH. A weak waterpump would definitely cause that. A waterpump has fins inside of it. As the engien runs, the belt turns the water pump pulley, which turns the fins that push the coolant through the system. As the car ages, sometimes those fins get a little bent and don't push with the same force as they used to. When you rev the engine, they're running faster and pushing more coolant through the system.

I figure it's either that, or there's a clog somewhere in the system, or the cylinder head and/or gasket is bad.

If the system is clogged somewhere, flushing it should help you out. Here's a how-to guide: https://www.2carpros.com/articles/coolant-flush-and-refill-all-cars

Also, finally, there could be mud/dirt/gunk caked onto the front of the radiator. Take a water hose and spray it off. If you can't see the front of the radiator (which I don't think you'll be able to on that car), just hose it down real good from in front of the grill. I kind of doubt this is the problem because the engine does cool when you rev it, but it's a free thing to do so it can't hurt to try it.

Personally, I'd start with the flush and spraying off the radiator. Then go for the water pump if flushing and cleaning didn't fix the problem.