It’s been awhile since I’ve tinkered with a fl250, but I do have two immediate thoughts.

1. Do you still have the original exhaust pipe? If so try it with the original pipe to eliminate that as a variable. Most likely that has nothing to do with your problem, but 2stroke exhaust design is critical to the Engine running properly, maybe your aftermarket one isn’t quite right.

2. Something is wrong with the ignition. I never had a CDI (Capacitor Discharge Ignition) (Capacitor Discharge Ignition) model, but I believe there is some adjustment, if the timing is off it would still spark, but at the wrong time and it won’t run correctly. You need a manual with the specifics of how to adjust the CDI (Capacitor Discharge Ignition) (Capacitor Discharge Ignition) for proper timing.

These are just two ideas, I could be completely off base, but if you have spark, compression (make sure it is at the proper level compression per the manual) and fuel it’s probably a timing issue. Lastly, what happens if you give it a shot of starter fluid. Does it start then? If yes, then timing probably isn’t the problem. Let us know what happens and we might be able to give you some more better advise.

Good luck!

I don't know how helpful I can be here because I'm NOT an electron mechanic LOL.
However I will attach a pdf manual.
On the second last page is a wiring diagram for the CDI (Capacitor Discharge Ignition) (Capacitor Discharge Ignition) but it does fire so nothing wrong here.
It's clearly an out of time issue.
Check the trigger coil, flywheel, make sure timing marks line up.
My money says timing marks not set right.
CO

Note: Site won't allow me to attach pdf because file to big. So you will have to go here to download a free manual: https://oddatv.com/fl250-fl350-factory-manuals/
They sell parts too.
CO

Thank you! I know it’s CDI (Capacitor Discharge Ignition) (Capacitor Discharge Ignition), but I never knew it could be adjusted. That’s new information for me. But by the looks of it I have the marks set right, but I’m going to take the flywheel off and check everything out again. I will keep you all updated, thank you!

it may just be a carb adjustment in the idle circuit, when you get it started are you able to keep it running using the throttle? are you using the choke?
I have found through the years that different engines like to start different ways, one thing in common, bring the Engine to TDC (Top Dead Center) (Top Dead Center) then just past so you can get it spinning good before coming back up on the compression cycle.
I have had engines that like to start on the choke and will run with the choke until warmed up, others that start on the choke when cold and immediately turn off the choke and continue to warm up using the throttle, some like the choke on for a couple of kicks, if it doesn't start then turn the choke off and start, others that only need the choke when the Engine is cold and cold out side. I'd say that a properly tuned machine will always act the same but I'm the sort that will sacrifice wrenching time for riding time :)
if it works and not burning up pistons and beats your buddy, why mess with it. I always ran my kids machines a little on the rich side.

CDI (Capacitor Discharge Ignition) (Capacitor Discharge Ignition) can NOT be adjusted. They're fixed.
But timing can be adjusted if the trigger coil is moved. That's dangerous though unless you know what you are doing.
Right now my money says the timing marks are not aligned or the flywheel key is sheared.
Kuma has good points here too. Are you sure your pilot jet is clean ??
CO

Edit: Is this a factory CDI (Capacitor Discharge Ignition) (Capacitor Discharge Ignition) oddy or did someone scab on a CDI (Capacitor Discharge Ignition) (Capacitor Discharge Ignition) aftermarket unit ??
CO

Below is a pic of a FL350 flywheel.
You will see that it has two marks. It has a "T" and an "F".
The "T" is for top and the "F" is for full advance. I think that's right.
With your piston at top dead center you want the "T" line and the other line to line up.
It's been a long time since I have F'd around with this.

In my opinion I would just verify that:
1) Your flywheel woodruff key is good and not sheared
2) Timing marks align
3) Are you actually getting fuel -- it may be no fuel flow and it's leaning out -- check clean jets and fuel pump flow. This "might" be the real problem because you have spark. If something was wrong with the timing like a sheared key then most likely you wouldn't have spark. Also if the FL250 is like the FL350 then the stator plate and trigger coil are fixed. There is NO adjustment because they just bolt in place. You would have to modify to get adjustment.
CO

Few more since you have it apart.

Since you working with the coil and wire.
Unscrew the plug cap and cut a 1/4 inch of wire off then screw back on. You can also do the same where it attaches to the coil. Then retest before you buy a new coil.
This is a common issue over time.
The plug wire get a high resistance from burnt or corroded connection.
This for the secondary test.

If this does make a difference let us know what your reading are for other to read and help others in the future.

One thing to check if you go forward with citing the wire is look at the connection points on the coil and cap. I have seam the caps tip Burton off and new cap needed. If you need a new cap get one that has a matching resistance built it.

In the pic showing the cap it says" noise. Suppressor" this is internal resistance built into cap.