[KDS4444]

Your offer to put me in touch with a person out here who could help me is very generous. There is nothing like a personal network connection to actually get things done! As it is, though, it seems like I am on my own for now.
I tested the spark plug, and had some surprises:
The plug tested out just fine as far as resistance goes, so I grounded it and cranked the kickstarter. It sparked! A bright blue-white spark came shooting right across the gap, just as it should! I was pleased, and put it back in the engine. This time, there was something like turnover happening. The engine didn't exactly catch and start turning over on its own, at least not at first, but I tried cranking it again a few more times and one of those times the engine actually ran for a second or two on its own!
But it didn't sound right. It's hard to put my finger on it, or describe the sound itself, but it didn't sound the way I know it should. It also wouldn't take off like that a second time for me. But it shows me that I am getting ignition. So my guess now (and probably yours as well) is compression. If I have a leak anywhere around the cylinder base, head, or spark plug itself, this is the kind of response I might get from the engine. "Yay, fuel! Spark! Ignition! Move that piston! Oh, wait, not enough compression.... Sorry! Goin' down!" I've had a compression issue once in the past, and it sounded like a giant dragon fly was inside my engine— it was the sound of compressed air escaping around the cylinder (oh and yes, quickly, I am sure the piston is facing the correct way, I was VERY careful about that! The arrow on the head of it points towards my exhaust port!). This didn't sound quite like that, but it could certainly still be the problem here.
Or, as you said, it might be the damn reed valve. That's one part I have not taken out and had a look at yet, esp. because it isn't terribly accessible. But as you said a cracked reed would also prevent me from getting proper compression in the cylinder. Are there tests I can do to see if my cylinder is leaking somewhere first? I can also try tightening my cylinder nuts down just a bit tighter and see if that helps at all. But I feel good knowing that at least I don't have an electrical problem and that my spark plug is working as it should.
Thank you for continuing to engage me here. I really appreciate the willingness to keep with me and help me try to solve this problem. That means a lot.
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Later that evening...
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Okay, I went down and tightened up the spark plug and cylinder nuts again, and gave the kickstart a go. The engine caught! I was amazed! It still didn't sound right, but it was running! I let it run for 3 or 4 seconds, and pondered taking it off the kick stand and driving it around the block. But then things suddenly changed: the speed of the piston in the cylinder picked up, and the sound coming from the engine became much more raspy and loud. I shut down the engine immediately. The sound was unlike anything I have ever heard before coming from the scooter.
It seems that I am finally getting compression, at least briefly, inside the cylinder. But the rasping noise? Would a cracked reed valve petal make a noise like that??
One more thing: I noticed when I put my air filter box back on that there were some cracks in the elbow joint where it attaches to the carburetor. I assume I will now need a new air filter box. The cracks mean that some unfiltered air is now able to get past the filter and go straight into the carburetor. Would this be causing the sounds I heard?