It DEFINITELY sounds like ignition. Anything mechanical should show up at almost any RPM. It sounds like, based on what you say, that the problem is happening when the engine is under a load, and at specific RPM ranges.
You said you replaced points and plugs. How about the condenser?
Check the wire coming from the distributor to the points. Does it have any cracks in the insulation?
Check the top of the coil. Is it really clean? No dust or oil? Could be arcing between the coil primary wire and one of the terminals on either side.
If you can suck on the vacuum advance and make it move, the vacuum advance is probably bad, because it takes one helluva lot of suction to make enough vaccum to cause it to move. I've always had to use a hand-held vacuum pump to create that much suction. (Or, go down to a local street corner and pick up a hooker!!! LOL) Possibly a broken spring in the vacuum advance unit.
Worn breaker plate should show up with a dwell meter when you rev the engine, whether the vacuum advance is connected or not. If the dwell is consistent when the engine is revved, look elsewhere.
Just because wires "look" good, that means nothing with resistance-type wires, which is probably what you have (I haven't seen solid-core wires in many years). The core can have too much or too little resistance, and will cause a plug to misfire under a load when the demand for spark is increased. Replace them!!!
Eliminate the cheap-to-fix possibilities before you go chasing the expensive stuff.
Actually, you'll come out ahead if you just go ahead and bite the bullet and find someone with an older engine analyzer with a scope. The cost to hook it up to one of these should be low, and you can isolate the problem in just a few minutes, rather than spending many hours and many dollars guessing.
The only other causes I can think of have already been mentioned. A bad cam lobe on the intake side would cause backfire through the carb, but that would sound like a popcorn popper, and would be consistent under wide open throttle at most any RPM under a load. Bad exhaust valve should cause a misfire (skip) at most any RPM. A simple vacuum test should show you a bad or leaky valve that would be leaking enough to cause a problem. Bad accelerator pump would cause severe hesitation and probably a single backfire when you first push the accelerator pump, even sitting still.