Often time can be saved on that kind of job by testing things in modules. The main advantage you have is that (apparently) it hasn't received any amateur attentions, so there is probably only one fault. We are being aimed in a particular direction by your comment that it has weak ignition. If you had not said that I'd be asking questions about compression: you said it is strong, but does it have strong crankcase compression as well as cylinder compression? A crankcase induction two stroke must have two compressions (as felt on the starter cord) per revolution.
If it has proper compression (two more or less equal, sharp resistance periods per revolution) the next issues to consider are fuel, spark, induction leaks, and exhaust blockages.
You said you had tried squirting a few drops of fuel into the spark plug hole, and it then kicked but didn't run. You need to put in a spoonful of fuel to do that test, or you may get the result that you got, from a perfectly healthy engine.
When you test spark it is essential to be testing the ignition system, not the spark plug. Hence if you do not see a steady stream of blue sparks across the plug, hold the plug lead about 3 mm from the cylinder and pull the cord. If the spark will not jump 3 mm it is pretty awful - a newish magneto should do better than 10 mm. I've had old Villiers engines with ignition about that weak that ran perfectly, but they were rather hard to start. Never condemn an ignition module without testing the direct spark - spark plugs, new or old, may be defective. I think I recall 4 instances here this year, where we went through long diagnostic procedures unnecessarily because a brand new spark plug would not spark.
One of the standard things to check in these cases is whether the muffler is blocked. This can happen in two main ways: wasp nests, and carbon build-up. The latter typically happens to two strokes that run for long periods, especially at light load or if the owner mixes the fuel with a heavy hand on the oil. It can also happen to four strokes with faulty piston rings.
In the end, I usually find the quickest and most effective way to check the entire induction system (but not the crankcase seals) is the fuel-through-the-plughole trick. If it runs for a few seconds on that spoonful, you have a running engine that has a fuel delivery or induction leak problem.
