I bought it with the installation in poor condition.If it's starting in gear I'd start checking the safety switches that stop it starting in gear, time to break out the test light & multi meter & study the wiring diagram, other things to consider are, when did this problem start, was I doing any maintenance to the atv at the time, a year model & ownership history will help.
the key is new and cuts off all the current except the starter motor. With the off switch cuts the sparkDoes turning the key on or off do anything?
If you unplug the module with the kill/start switch, does the key on/off do anything?
There are definitely some on here with 660 experience, I'm not one of them so the above is just guessing to what is similar on my 700.
Start unpluging the switches and test them per the 660 service manual.