FWIW, your piston video looks fine to me. The damper has a one-way valve in it, so the piston is supposed to fall much quicker than it rises. And you should be able to get it to fire pretty much no matter how badly the carbs are worn. For cold starting, the jets should be lowered quite a bit, not against the mixture nut (so adjusting the nut should have no effect).
Two things that often get recalcitrant engines started for me :
1) Take the plugs out and heat the tip with a propane torch, until there is no more yellow in the flame. Any yellow is carbon burning off the tip of the plug, and carbon can present a path to short out the spark. Obviously hold the plug with pliers or similar during the process.
2) Shoot a healthy dose of spray carburetor cleaner (rather than "starting fluid") down each carb throat and then immediately crank the engine. If it runs for a second or two and then dies; you've got some kind of fuel delivery problem. Could be a blocked fuel passage, or bad fuel in the carbs.
But if it won't even sputter, more likely it's an ignition problem. It takes more voltage to fire a plug under compression, so having a spark jump the gap outside the engine doesn't always mean the ignition is fine. As noted, timing is important too. If all else fails, double-check that you're getting a spark when the cylinder is at (near) TDC between compression and power strokes. As I learned the hard way, they won't run if the spark is at TDC between intake and exhaust