As above... the clear filter will look empty, but fuel will flow. Basicaly, air can get trapped behind the paper.
Lets go down the list
1) If you have an air leak... that's BAD. A lean run on a 2-stoke will kill the engine. SO, double check ALL the hoses, and make sure they are clamped.
2) These engiens are sensitive to voltage. SO, a low battery can casue issues. BUT, if it can crank and start... it's probably OK. But, put a meter on the battery, and check it, a) Sitting, b) Cranking, c) at idle/running.
3) As above, check the fuel pressure. DO NOT GUESS.
4) You said you have spark on all 6, but then said you want to check the trigger. ?? Here again... DO NOT GUESS. verify, 100% that you have spark on all 6. Get 3 in-line testers (They are cheap) plug them in... start the engine, and simply look for a nice blue spark. It was suggested to pull the wires while running... but I don't recommend that. a) You can shock yourself. b) These engines idle just fine with a couple wires off. c) If you get a good spark, while removing it... you could cause an explosion from bilge fumes !!!!!!!!! You may get away with it on a 2/3 cyl ski, but not on a 6 cyl boat, and bilge explosions are real.
5) Check the compression. Post the numbers for us.
6) Did you verify that the throttle plates, at the engine are opening all the way?
A smoky 240 is absolutely normal at low RPM. Especially after it sits. Mine will fog out the dock when I first put it in the water, if it's been sitting for a week. Even after a long idle/low RPM ride... it will be wafting white smoke. It's simply how a 2-stroke is. With that said, a bad port side temp sender, or bad TPS will make it worse, but you will also have starting issues, and low RPM at idle. (Target idle is 1100 RPM in the water)