It sounds like you crank the engine but the engine won't start. Do you use the choke? Mine won't start without it. I pull it half way out then it starts first go. Then push it back in after a minute or two. This only applies to the first cold start. Once warmed up it doesn't need choke to restart. Even in summer it just cranks away without starting without the choke.
If your battery is not in good condition or getting a bit old (maybe a few years) you will quickly flatten the battery during cranking. i don't think it's the cdi or else you wouldn't be able to push start it. The engine simply can't run without the cdi as it provides the spark. If your battery is getting flat over night though, or not holding charge, you will be able to push start it as you apparently can, but it won't start the next day with the electric starter button (or perhaps even if you stop the engine and try to restart it.).
Does the generator charge light stay on after you push start it? If so the generator may not be charging the battery and therefore you will have a flat battery. In any case if the generator light comes on while you are riding then it suggests a fault in the charging system. eg battery/ generator/ wiring to either. Also, if you are cranking but don't see spark it could be that the cdi isn't getting the right volts/current . Put a multimeter across the battery terms with IGNITION OFF, you should see 12.5 to about 13 volts. If it's any less than 12v the battery is probably not charging properly or not holding charge. My guess is that the cdi needs about 12v minimum to generate sufficient spark to start the engine. And don't charge your batt with a car charger as you will likely fry the bike battery. If you already did use a car batt charger, you may have damaged the bike batt.
