Did you remove the main jet and clean it so you can see through the hole?

Do you have both the ignition key and the RUN siwtch on the right handle bar on?

Are you pushing down on the choke lever on the back of the carb each time you try to start it?

Does it have the air filter box on the back of the carb?

The screw on the side is for the idle adjustment, it isn't an air mixture screw. You screw it in to increase idle speed and out to decrease.

Could the plug boot / wire be coming loose when you put the plug back in?

What about timing, does it appear to be in the ballpark? That spark needs to happen at the right time...

Good luck,
Sean

Doesn't sound like electrical problem if you have spark when the key and switch are turned on and no spark when your turn off the handle bar switch.

Are you kick starting or pedaling when it is on the center stand.

Try this;

All swicthes ON
1.get a new plug
2. Make sure the petcock is in a position where you have gas flow
3.squirt some starter fluid spary in the spark hole and out in the new plug
4. choke down on back of the carb and air cleaner box on the carb
5. get on and push the decompression lever on the left handle bar
6. pedal like like hell and give it some throttle.

You'll have to push down on the choke on each start try.

Sometimes it takes a lot of pedaling to get enough suction from the engine to pull gas throughy the carb on an old bike that has been setting for a long time.

Also check your tail light/brake light dual bulb to make sure it is not burned out.

now, here's a little test. spray carb cleaner into the carb throat with everything bolted back on (exhaust) and try to start it. if it fires, you have a fuel problem. if it does not fire, you have either an ignition or compression problem. and since you know it has spark, i would think ignition is good. timing isnt really an issue on these engines. they will fire no matter how far advanced or retarded the stator is. only a sheared flywheel key would be an issue.
make sure the head bolts are torqued to 10 ft/lbs. the screw on the carb is an idle speed adjustment, not a mixture screw. dont worry about where it is set until the engine is running. once running, you adjust this screw so that the engine runs at the highest speed possible without the rear wheel spinning while on the center stand.
the reasons it wants to start with the exhaust off is;
low compression
bad crankseal
without numbers i cant tell you which direction to go here. ive never seen a muffler so badly plugged that an engine wont at least fire and run for a couple seconds. they have to build up pressure in order for that to happen. think about a balloon, it take a bit to fill it to capacity right? well it takes a muffler a while to fill to capacity as well. and in most cases a plugged muffler affects top end more than idle speeds and starting.
while you have the exhaust off, look up into the exhaust port, and roll the engine over by hand and look at the piston side wall. if you see scratches or smear marks, you need to rebuild the top end. if you cant see the piston, clean the exhaust port out. while looking up at the piston, move it into position where you can see the rings. then use a prick punch, push inward on the rings to see if they move freely. if not, you need to rebuild the top end. if they move, you need to do a compression test to see how bad they are.
another idea for you is to take the bike to a lawn and garden repair shop, or a motorcycle shop, and have them do a leak down test to see where your compression is going.

i still think you have a dirty carb tho. or lowcompression?

donp

One important note on any older moped or bike (especially mopeds since the materials are so small and flimsy) be aware of welding joints. My right fork axle mount seperated from the fork while I was riding. Damn near killed me when it flipped me face first into the pavement and then landed on top of me.

None the worse for the crash I replaced the fork and ride still.